qu.1.topic=Normal Distribution Mathematic@

qu.1.1.mode=True False@
qu.1.1.name=27. Sample vs Population SD@
qu.1.1.comment=<p>The standard deviation of sample means <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&sigma;</mi><mrow><mover><mrow><mi>x</mi></mrow><mi>_</mi></mover></mrow></msub></mrow><mrow></mrow></mstyle></math> is related to the population standard deviation by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&sigma;</mi><mrow><mover><mrow><mi>x</mi></mrow><mi>_</mi></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>&sigma;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>. $EX Thus the statement is false.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$Q="27";
$Pick=rint(3);
$X = switch($Pick,"The standard deviation of sample means is larger than the standard deviation of the population, it also equals the population standard deviation multiplied by the square root of the sample size.","The standard deviation of sample means is smaller than the standard deviation of the population, it also equals the population standard deviation multiplied by the square root of the sample size.","The standard deviation of sample means is larger than the standard deviation of the population, it also equals the population standard deviation divided by the square root of the sample size.");
$EX = switch($Pick,"The standard deviation of sample means cannot be larger than the standard deviation of the population. Also it does not equal the population standard deviation multiplied by the square root of the sample size.","The standard deviation of sample means does NOT equal the population standard deviation multiplied by the square root of the sample size.","The standard deviation of sample means cannot be larger than the standard deviation of the population.");@
qu.1.1.uid=a92ccb70-a8cc-4124-8691-53304897cca3@
qu.1.1.info=  Type=TF;
  Course=202;
@
qu.1.1.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q">$X</div>@
qu.1.1.answer=2@
qu.1.1.choice.1=True@
qu.1.1.choice.2=False@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=28b. P(Mean>z)@
qu.1.2.comment=<p>Let the mean of the sample be X. Then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$Q="28b";
$N = range(100,110);
$U = range(40,45);
$S = range(25,30);
$X = range($U+1,$U+3);
$Z = decimal(4,($X-$U)/($S/sqrt($N)));
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$Ans = decimal(4,1-$P);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+$Alt1));
$Alt4=decimal(4,0.5*($Ans+$Alt2));@
qu.1.2.uid=1a6e545a-24ad-49be-b9e2-15548c27eaf6@
qu.1.2.info=  Course=202;
  Type=MC;
@
qu.1.2.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q">A random sample of&nbsp;$N observations is to be drawn from a population with a mean of&nbsp;$U and a standard deviation of $S. The probability that the mean of the sample will exceed&nbsp;$X is:</div>@
qu.1.2.answer=1@
qu.1.2.choice.1=$Ans@
qu.1.2.choice.2=$Alt1@
qu.1.2.choice.3=$Alt2@
qu.1.2.choice.4=$Alt3@
qu.1.2.choice.5=$Alt4@
qu.1.2.fixed=4@

qu.1.3.mode=Multiple Choice@
qu.1.3.name=08. Given SD &  P(Z < x): mean is?@
qu.1.3.comment=<p>With
<title></title>
<meta content="Microsoft FrontPage 5.0" name="GENERATOR" />
<meta content="FrontPage.Editor.Document" name="ProgId" />&mu; as the mean, standardize the variable to get a Standard Normal <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mrow></mstyle></math> .</p>
<p>So <font size="3" face="Times New Roman"><em>P</em>(<em>X</em> < $X) =</font> <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math> <font size="3" face="Times New Roman">= $P</font> (given) . Use the inverse normal table or calculator to find:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$X</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$S</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$I</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>&nbsp;</p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$Q=8;
$S = range(1,4,0.001);
$X = range(7,8,0.001);
$P = range(0.5,0.7,0.0001);
$PreI = maple("(stats[statevalf,icdf,normald])($P)");
$I=decimal(4,$PreI);
$Ans = decimal(4,$X - $S*$I);
$Alt1 = decimal(4,$Ans*range(1.1,1.5,0.01));
$Alt2 = decimal(4,$Ans*range(0.4,0.9,0.01));
$Alt3 = decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.3.uid=7f118863-35c6-4253-8ded-18732df7c7eb@
qu.1.3.info=  Course=202;
  Type=MC;
@
qu.1.3.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>If X is a normal random variable with standard deviation $S, and if the probability that X is less than $X is $P (as shown below), then what is the mean of X? (Note: the diagram is not necessarily to scale.)<br />
<img width="287" height="195" src="__BASE_URI__CPD/ND/NormalLeft70PC.gif" alt="Normal curve, 70% point." title="Normal curve, 70% point. [IMG:NormalLeft70PC.gif]" /></div>@
qu.1.3.answer=1@
qu.1.3.choice.1=$Ans@
qu.1.3.choice.2=$Alt1@
qu.1.3.choice.3=$Alt2@
qu.1.3.choice.4=$Alt3@
qu.1.3.fixed=@

qu.1.4.mode=Inline@
qu.1.4.name=04. Describe the Distribution@
qu.1.4.comment=<p>This is an exact Normal distribution. In this case the sample mean is the same as the population mean, and the sample standard deviation is: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>&sigma;</mi><mrow><mi>X</mi></mrow></msub><mrow><msqrt><mrow><mi>N</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SDAns</mi></mrow></mstyle></math></p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$Q=4;
$Which=1+rint(6);
$Align=switch(rint(2),"Left","Right");
$Mu=decimal(1,range(6.0,24.0,.1));
$SD=decimal(1,range($Mu/30,$Mu/10,0.1));
$SDAns=decimal(2,$SD/2);
$FakeSD1 = $SD;
$FakeSD2 = decimal(2,($SD + $SDAns)/2);@
qu.1.4.uid=92da1826-630d-4c53-9d18-19239b7b56fc@
qu.1.4.info=  Course=202;
  Type=MC;
@
qu.1.4.weighting=1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.fixed=@
qu.1.4.part.1.choice.4=approximate normal distribution with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi></mrow></mstyle></math> = $Mu, s = $FakeSD2<br>@
qu.1.4.part.1.question=null@
qu.1.4.part.1.choice.3=exact normal distribution with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi></mrow></mstyle></math> = $Mu, s = $SDAns<br>@
qu.1.4.part.1.choice.2=exact normal distribution with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi></mrow></mstyle></math> = $Mu, s = $FakeSD1@
qu.1.4.part.1.choice.1=approximate normal distribution with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi></mrow></mstyle></math> = $Mu, s = $FakeSD1@
qu.1.4.part.1.mode=Multiple Choice@
qu.1.4.part.1.display=vertical@
qu.1.4.part.1.answer=3@
qu.1.4.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__CPD/ND/Cookie$Which.gif" alt="Cookie" title="Cookie [IMG:Cookie$Which]" />The weights of packets of a certain type of cookie follow a normal distribution with mean $Mu grams and standard deviation $SD grams. The average weight of a simple random sample of four packets of cookies then has an<span> </span><1><span> </span></div>@

qu.1.5.mode=Inline@
qu.1.5.name=25. Continuity Correction@
qu.1.5.comment=@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=@
qu.1.5.uid=813bdc60-8d60-4786-9ca3-1a0d33145325@
qu.1.5.info=  Type=FITB;
@
qu.1.5.weighting=1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.grader=relaxed@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.answer.3=continuity corrections@
qu.1.5.part.1.answer.2=continuity correction@
qu.1.5.part.1.answer.1=correction for continuity@
qu.1.5.part.1.mode=List@
qu.1.5.part.1.display=text@
qu.1.5.part.1.credit.3=1.0@
qu.1.5.part.1.credit.2=1.0@
qu.1.5.part.1.credit.1=1.0@
qu.1.5.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q25">A(n)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> is employed when a continuous distribution is used to approximate a discrete distribution.</div>@

qu.1.6.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>Suppose X has a normal distribution with mean $MeanIs and variance $V. <br />
<br />
Find the constant <font size="3" face="Times New Roman"><em>c</em></font> so that <font size="3" face="Times New Roman"><em>P</em>(|<em>X</em> - $MeanIs| &le; <em>c</em>) = $P&nbsp; </font>(3 decimals)</div>@
qu.1.6.answer.num=$Ans@
qu.1.6.answer.units=@
qu.1.6.showUnits=false@
qu.1.6.grading=toler_abs@
qu.1.6.err=0.01@
qu.1.6.negStyle=minus@
qu.1.6.numStyle=thousands scientific dollars arithmetic@
qu.1.6.mode=Numeric@
qu.1.6.name=09. μ, σ² & P(|x-#|≤c): Find c@
qu.1.6.comment=<p>We have a variance <font size="3" face="Times New Roman"><em>&sigma;<sup>2</sup> = </em>$V</font>, a mean <font size="3" face="Times New Roman"> <em>&mu; = </em>$MeanIs</font>.</p>
<p>Express this question in a <em>standardized Normal</em> form:</p>
<p>We want <em>c</em> such that<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><mi mathvariant='normal'>$c</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$V</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi mathvariant='normal'>$MeanIs</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$V</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mrow><mfrac><mi mathvariant='normal'>$c</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$V</mi></mrow></msqrt></mrow></mfrac></mrow></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><mi>c</mi><mrow><mi mathvariant='normal'>$StdDevIs</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mi>c</mi><mrow><mi mathvariant='normal'>$StdDevIs</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.8</mn></mrow></mstyle></math></p>
<p>Then this can be expressed as</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mi>c</mi><mrow><mi mathvariant='normal'>$StdDevIs</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><mi>c</mi><mrow><mi mathvariant='normal'>$StdDevIs</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mi>c</mi><mrow><mi mathvariant='normal'>$StdDevIs</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mi>c</mi><mrow><mi mathvariant='normal'>$StdDevIs</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mi>c</mi><mrow><mi mathvariant='normal'>$StdDevIs</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$P</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p>Then use the Inverse Normal Calculator to find:</p>
<p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>c</mi><mrow><mi mathvariant='normal'>$StdDevIs</mi></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$fPreAns</mi></mrow></mstyle></math>,</p>
<p><font size="3" face="Times New Roman"><em>c</em> = $fPreAns*$StdDevIs = $Ans </font>.</p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$Q=9;
$P=decimal(2,range(0.80,0.99,.01));
$V=decimal(2,range(1,6,0.5));
$MeanIs=range(1,10,1);
$StdDevIs=decimal(3,sqrt($V));
$PreAns=maple("(stats[statevalf, icdf, normald])((1+$P)/2)");
$fPreAns=decimal(3,$PreAns);
$Ans=decimal(3,$StdDevIs*$PreAns);@
qu.1.6.uid=c9ad621b-f668-4b29-a3e5-68506b18b773@
qu.1.6.info=  Course=230;
  Type=numeric;
@

qu.1.7.mode=Multiple Choice@
qu.1.7.name=21. σ, P(X > x) : what is μ?@
qu.1.7.comment=<p>Standardize, so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>&mu;</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> = $P or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mover accent='true'><mo lspace='0.0em' rspace='0.1111111em'>&minus;</mo><mi></mi></mover><mi mathvariant='normal'>&mu;</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z</mi></mrow></mstyle></math></p>
<p>Where &mu; is the mean (of X). Use the inverse normal to find: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>&mu;</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$I</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> so  &mu; = $X - $S*($I) = $Ans</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$Q=21;
$S = range(2,2.5,0.1);
$X = range(5,7,0.001);
$P = range(0.1,0.7,0.0001);
$Z = 1-$P;
$PreI = maple("(stats[statevalf,icdf,normald])($Z)");
$I=decimal(4,$PreI);
$Ans = decimal(4,$X - $S*$I);
$Alt1=decimal(4,range(1.1,1.9,0.01)*$Ans);
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.7.uid=7502e424-62b3-4e02-a18b-0c0d4154255c@
qu.1.7.info=  Course=202;
  Type=MC;
@
qu.1.7.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>If X is a normal random variable with standard deviation $S, and if the probability that X is more than $X is $P (as shown below), then what is the mean of X? (Note: the diagram is not necessarily to scale.)
<p>&nbsp;</p>
<p><img width="289" height="176" align="middle" src="__BASE_URI__CPD/ND/NormalRightTail.gif" alt="Normal graph with right tail." title="Normal graph with right tail. [IMG:NormalRightTail.gif]" /></p>
</div>@
qu.1.7.answer=1@
qu.1.7.choice.1=$Ans@
qu.1.7.choice.2=$Alt1@
qu.1.7.choice.3=$Alt2@
qu.1.7.choice.4=$Alt3@
qu.1.7.fixed=@

qu.1.8.mode=Multiple Choice@
qu.1.8.name=11. μ, P(X < x):  What is SD?@
qu.1.8.comment=<p>For <font size="3" face="Times New Roman"><em>Z</em> ~ <em>N</em>(0,1),&nbsp; <em>P</em>(<em>Z</em> < <em>z</em>) = $P</font> means <font size="3" face="Times New Roman"><em>z</em> = $I</font> (do an inverse table lookup, or use the calculator provided).</p>
<p>Standardizing our r.v. <em><font size="3" face="Times New Roman">X</font></em> we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&mu;</mi></mrow></mrow><mrow><mi>SD</mi></mrow></mfrac></mrow></mrow></mstyle></math> so <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi>SD</mi></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>and solving:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$I</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$Q=11;
$U = range(3,5,1);
$X = range(5,6,0.001);
$P = range(0.6,0.7,0.001);
$PreI = maple("(stats[statevalf,icdf,normald])($P)");
$I=decimal(4,$PreI);
$Ans = decimal(4,($X - $U)/$I);
$Alt1 = decimal(4,$Ans*range(1.1,1.5,0.01));
$Alt2 = decimal(4,$Ans*range(0.4,0.9,0.01));
$Alt3 = decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.8.uid=35bfcd92-7c92-4b24-91ce-7ae5bbdf2122@
qu.1.8.info=  Course=202;
  Type=MC;
@
qu.1.8.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>If X is a normal random variable with mean $U, and if the probability that X is less than $X is $P (as shown below), then what is the standard deviation of X? (Note: the diagram is not necessarily to scale.)
<p>&nbsp;</p>
<img width="287" height="195" src="__BASE_URI__CPD/ND/NormalLeft70PC.gif" title="Normal Distribution, showing 70% point [IMG:NormalLeft70PC.gif]" alt="Normal Distribution, showing 70% point" /></div>@
qu.1.8.answer=1@
qu.1.8.choice.1=$Ans@
qu.1.8.choice.2=$Alt1@
qu.1.8.choice.3=$Alt2@
qu.1.8.choice.4=$Alt3@
qu.1.8.fixed=@

qu.1.9.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>What is the z value to the right of the mean such that $Ppc % of the total area lies to the left of it as shown in the figure below?&nbsp; Diagram may not be to scale. (3 decimal accuracy please.)
<p>&nbsp;</p>
<p><img width="267" height="184" src="__BASE_URI__CPD/ND/NormalLeft85PCV2.gif" alt="Normal Curve, 85% point." title="Normal Curve, 85% point. [IMG:NormalLeft85PCV2.gif]" /></p>
</div>@
qu.1.9.answer.num=$Ans@
qu.1.9.answer.units=@
qu.1.9.showUnits=false@
qu.1.9.grading=toler_abs@
qu.1.9.err=0.01@
qu.1.9.negStyle=minus@
qu.1.9.numStyle=thousands scientific dollars arithmetic@
qu.1.9.mode=Numeric@
qu.1.9.name=07. z such that P(Z < z) = p@
qu.1.9.comment=<p>Just find the Inverse Normal for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$Ppc</mi></mrow><mrow><mn>100</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$Q=7;
$p=range(0.75,0.89,0.01);
$Ppc=100*$p;
$PreAns=maple("(stats[statevalf, icdf, normald])($p)");
$Ans=decimal(3,$PreAns);@
qu.1.9.uid=87f382ef-19fa-4cb6-93bf-66d7371359f4@
qu.1.9.info=  Course=202;
  Type=numeric;
@

qu.1.10.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>Let <font size="3" face="Times New Roman"><em>Z</em> ~ <em>N</em>(0,1)</font> . Find a <font size="3" face="Times New Roman"><em>Z<sub>0</sub></em></font> such that the area to the right of <font size="3" face="Times New Roman"><em>z</em> = Z<sub>0</sub>&nbsp; </font>is <font size="3" face="Times New Roman">$RightArea</font>. (3 decimals)</div>@
qu.1.10.answer.num=$Ans@
qu.1.10.answer.units=@
qu.1.10.showUnits=false@
qu.1.10.grading=toler_abs@
qu.1.10.err=.01@
qu.1.10.negStyle=minus@
qu.1.10.numStyle=thousands scientific dollars arithmetic@
qu.1.10.mode=Numeric@
qu.1.10.name=06. Find z for given area of Z > z@
qu.1.10.comment=<p>First calculate the area <em>to the left</em> of <em><font size="3" face="Times New Roman">Z<sub>0</sub></font> :</em>&nbsp; <font size="3" face="Times New Roman"><em>A</em> = 1 - $RightArea = $LeftArea.</font>&nbsp;</p>
<p>This means <font size="3" face="Times New Roman"><em>P</em>(<em>z</em> < Z<sub>0</sub>) = $LeftArea</font>.</p>
<p>Do a reverse lookup of this value in a Normal table, or use the calculator provided to find <font size="3" face="Times New Roman"><em>Z<sub>0</sub></em> = $Ans</font></p>@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$Q=6;
$Ans=range(0.1,1.92,0.01);
$PreLA=maple("(stats[statevalf, cdf, normald])($Ans)");
$LeftArea=decimal(4,$PreLA);
$RightArea=1-$LeftArea;@
qu.1.10.uid=8b07dfaa-3bff-42ff-a914-e97f842a5bf4@
qu.1.10.info=  Course=230;
  Course=202;
  Type=numeric;
@

qu.1.11.question=<div title="University of Waterloo Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>If X is a normal random variable with mean $U and standard deviation $SD, then find (3 decimals) the value x such that P(X > x) is equal to $P, as shown below. (Note: the diagram is not necessarily to scale.)
<p>&nbsp;</p>
<p><img width="294" height="196" title="Normal graph, 70% [IMG:Normal70PC.gif]" src="__BASE_URI__CPD/ND/Normal70PC.gif" alt="" /></p>
</div>@
qu.1.11.answer.num=$Ans@
qu.1.11.answer.units=@
qu.1.11.showUnits=false@
qu.1.11.grading=toler_abs@
qu.1.11.err=0.01@
qu.1.11.negStyle=minus@
qu.1.11.numStyle=thousands scientific dollars arithmetic@
qu.1.11.mode=Numeric@
qu.1.11.name=23a. P(Z > x), find x@
qu.1.11.comment=<p>P(X > x) = $P means 1 - P(X < x) = $P so P(X < x) = 1 - $P = $Z&nbsp;</p>
<p>Standardizing:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z</mi></mrow></mstyle></math> so using the Inverse Normal we have :</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$I</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$I</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$U</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.11.editing=useHTML@
qu.1.11.solution=@
qu.1.11.algorithm=$Q="23a";
$U = range(5,9,1);
$SD = range(2,3,0.1);
$P = range(0.7,0.8,0.0001);
$Z = 1-$P;
$PreI = maple("(stats[statevalf,icdf,normald])($Z)");
$I=decimal(4,$PreI);
$Ans = decimal(4,$U + $SD*$I);@
qu.1.11.uid=26bd86e6-193a-442a-a482-b9bb8313d52e@
qu.1.11.info=  Course=202;
  Type=numeric;
@

qu.1.12.mode=Multiple Choice@
qu.1.12.name=16. Distance from the mean@
qu.1.12.comment=<img width="289" hspace="4" height="176" align="right" src="__BASE_URI__CPD/ND/NormalCurve2.gif" alt="Normal Curve" title="Normal Curve [IMG:NormalCurve2.gif]" />Recall the normal distribution's shape. The farther you are from the mean, the lower the frequency (probability of occurrence). $ans is $Gap from the mean of $M, the most of any of the choices.</p>@
qu.1.12.editing=useHTML@
qu.1.12.solution=@
qu.1.12.algorithm=$Q=16;
$M=range(20,50,1);
$Gap=range(6,24,2);
$SD=range(3,8,1);
$SignIs=switch(rint(2),-1,1);
$ans=$M+$SignIs*$Gap;
$alt1=$M+$SignIs*$Gap/2;
$alt2=$M-$SignIs*$Gap/2+1;
$alt3=$ans-$SignIs*range(1,abs($M-$ans)-2,1);
$alt4=$M-$SignIs*range(1,abs($M-$ans)-2,1);@
qu.1.12.uid=14eef2e4-0e25-47e9-964a-3cab2fa1e76e@
qu.1.12.info=  Course=230;
  Type=MC;
@
qu.1.12.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q">In a normal distribution of data, the mean is $M, and the standard deviation is $SD. Among the following, which raw score has the lowest frequency?</div>@
qu.1.12.answer=1@
qu.1.12.choice.1=$ans@
qu.1.12.choice.2=$alt1@
qu.1.12.choice.3=$alt2@
qu.1.12.choice.4=$alt3@
qu.1.12.choice.5=$alt4@
qu.1.12.fixed=@

qu.1.13.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>Suppose X and Y are independent random variables with X ~ N($Mx,$VarxIs) and Y ~ N($My,$VaryIs). Find the probability that X > Y (4 decimals please).</div>@
qu.1.13.answer.num=$Ans@
qu.1.13.answer.units=@
qu.1.13.showUnits=false@
qu.1.13.grading=toler_abs@
qu.1.13.err=0.001@
qu.1.13.negStyle=minus@
qu.1.13.numStyle=thousands scientific dollars arithmetic@
qu.1.13.mode=Numeric@
qu.1.13.name=24. P(X>Y) normals@
qu.1.13.comment=<p><strong>Correct answer: $Ans</strong></p>
<p>P(X > Y) = P(Y - X < 0). Y - X has a Normal Distribution with mean $My - $Mx = $M and variance = $VaryIs + $VarxIs = $VarIs.<br />
<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>Y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>Y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi>X</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$M</mi></mrow></mfenced></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$VarIs</mi></mrow></msqrt></mrow></mfrac><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mi mathvariant='normal'>&uminus0;$M</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$VarIs</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$SLimit</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$Q=24;
$Mx=range(4,16,2);
$My=range($Mx,$Mx+8,2);
condition:gt($My,$Mx);
$VarxIs=range(2,$Mx/2,1);
$VaryIs=range(2,$My/2,1);
$M = $My-$Mx;
$VarIs = $VarxIs + $VaryIs;
$StdDevIs=decimal(3,sqrt($VarIs));
$SLimit = -$M/$StdDevIs;
$PreAns = maple("(stats[statevalf, cdf, normald])($SLimit)");
$Ans=decimal(4,$PreAns);@
qu.1.13.uid=c4a36df4-f576-4d87-aefa-b9b866b7eea7@
qu.1.13.info=  Course=230;
  Type=numeric;
@

qu.1.14.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>Consider the standard normal Z ~ N(0,1). What <u>$SayWhich</u> z&nbsp; value has $Percent% of the data lying between itself and the mean? (4 decimal accuracy)</div>@
qu.1.14.answer.num=$Ans@
qu.1.14.answer.units=@
qu.1.14.showUnits=false@
qu.1.14.grading=toler_abs@
qu.1.14.err=.001@
qu.1.14.negStyle=minus@
qu.1.14.numStyle=thousands scientific dollars arithmetic@
qu.1.14.mode=Numeric@
qu.1.14.name=15a. Given Area(μ - z) = x%,  z is?@
qu.1.14.comment=<p>You know that (with the standard Normal) P(Z &le; 0) = 0.5 .&nbsp; There are <u>two</u> possible z values with $Percent% of the data lying between them and the mean, however you just need find one of them. You can do an inverse lookup in a table or use the calculator provided to obtain the positive z value with this property. $NegExp1</p>
<p>Consider z with P(Z &le; z) = 0.5 + $P&nbsp; . Use the Inverse Normal to find z = $AAns. $NegExp2</p>@
qu.1.14.editing=useHTML@
qu.1.14.solution=@
qu.1.14.algorithm=$Q="15a";
$WhichWay=rint(2);
$SayWhich=switch($WhichWay,"positive","negative");
$Sign=(-1)^$WhichWay;
$Percent = range(15,25,1);
$P = $Percent/100;
$Z = 0.5 + $P;
$I = maple("(stats[statevalf,icdf,normald])($Z)");
$Ans = $Sign*decimal(4,$I);
$AAns=abs($Ans);
$NegExp1=switch($WhichWay,"","The answer you need is just the negative of this (by the symmetry of the Normal distribution).");
$NegExp2=switch($WhichWay,"","The answer is the negative of this: $Ans.");
$Alt1=decimal(4,range(1.1,1.9,0.01)*$Ans);
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.14.uid=1edf797c-e8b5-42ae-a857-a5e21f4fd935@
qu.1.14.info=  Course=202;
  Type=numeric;
@

qu.1.15.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q">A normal population (represented by the r.v. X) has mean $Mean and standard deviation $SD. For a random sample of size $SS, determine the mean of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mover accent='false'><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>X</mi></mrow><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo></mrow></mover></mrow></mrow></math>. (3 decimal accuracy)</div>@
qu.1.15.answer.num=$Ans@
qu.1.15.answer.units=@
qu.1.15.showUnits=false@
qu.1.15.grading=exact_value@
qu.1.15.negStyle=minus@
qu.1.15.numStyle=thousands scientific dollars arithmetic@
qu.1.15.mode=Numeric@
qu.1.15.name=17a. Mean of Sample@
qu.1.15.comment=<p>The expected value of the mean of samples of any size is just the distribution's mean.</p>@
qu.1.15.editing=useHTML@
qu.1.15.solution=@
qu.1.15.algorithm=$Q="17a";
$SS=range(5,12);
$Mean=range(20,50);
$SD=range(3,$Mean/2);
$Ans=$Mean;@
qu.1.15.uid=b14d6136-30ad-4dd8-97fb-0db885b86430@
qu.1.15.info=  Course=202;
  Type=numeric;
@

qu.1.16.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q">


<a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=150,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>Let <font size="3" face="Times New Roman"><em>Z</em> ~ <em>N</em>(0,1)</font>. Find <font size="3" face="Times New Roman"><em>P</em>(<em>Z</em> &le; $z)</font>&nbsp;&nbsp; (3 decimal accuracy)</div>@
qu.1.16.answer.num=$Ans@
qu.1.16.answer.units=@
qu.1.16.showUnits=false@
qu.1.16.grading=toler_abs@
qu.1.16.err=0.01@
qu.1.16.negStyle=minus@
qu.1.16.numStyle=thousands scientific dollars arithmetic@
qu.1.16.mode=Numeric@
qu.1.16.name=01. Find P(Z ≤ #)@
qu.1.16.comment=This is quite simple - merely plug $z into the Normal calculator or look it up in a Normal table.@
qu.1.16.editing=useHTML@
qu.1.16.solution=@
qu.1.16.algorithm=$Q=1;
$z=decimal(2,range(-2,2,0.01));
$PreAns=maple("(stats[statevalf, cdf, normald])($z)");
$Ans=decimal(4,$PreAns);@
qu.1.16.uid=f8bc94f5-1f99-42bc-9951-c11ba88a9b25@
qu.1.16.info=  Course=230;
  Type=numeric;
@

qu.1.17.mode=Multiple Choice@
qu.1.17.name=03. P(z1 < Z < z2)@
qu.1.17.comment=<p><font size="3" face="Times New Roman"><em>P</em>($z1 < <em>z</em> < $z2) = <em>P</em>(<em>z</em> < $z2) - <em>P</em>(<em>z</em> < $z1) = <em>F</em>($z2) - <em>F</em>($z1) = $Fz2 - $Fz1 = $Ans</font></p>@
qu.1.17.editing=useHTML@
qu.1.17.solution=@
qu.1.17.algorithm=$Q=3;
$z1=range(-1.35,-1.05,0.01);
$z2=range($z1+0.2,1.26,0.01);
$C1="*****Maple: Find F(z2), F(z1), return as list L=[F(z2),F(z1)]*****";
$L=maple("A:=(stats[statevalf, cdf, normald])($z2):
B:=(stats[statevalf, cdf, normald])($z1):
[A,B]");
$Fz1=decimal(4,switch(1,$L));
$Fz2=decimal(4,switch(0,$L));
$Ans=$Fz2-$Fz1;
$Alt1=decimal(4,$Ans/2+range(-$Ans/4,$Ans/4,0.0001));
$Alt2=decimal(4,(1+$Ans)/2);
$Alt3=decimal(4,range($Ans+0.0002,$Alt2-0.0002,0.0002));@
qu.1.17.uid=525bf237-9fb1-46ad-a2ae-5c14f3ccca79@
qu.1.17.info=  Course=230;
  Type=MC;
@
qu.1.17.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator2.htm" onclick="window.open(this.href,this.target,'height=365,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>Find the probability <font size="3" face="Times New Roman"><em>P</em>($z1 < <em>z</em> < $z2)</font> using the standard normal distribution.</div>@
qu.1.17.answer=1@
qu.1.17.choice.1=$Ans@
qu.1.17.choice.2=$Alt1@
qu.1.17.choice.3=$Alt2@
qu.1.17.choice.4=$Alt3@
qu.1.17.fixed=@

qu.1.18.mode=True False@
qu.1.18.name=12. z the nth percentile ? (False)@
qu.1.18.comment=<p>"z corresponding to the $Percentile$Ending percentile" just means that P(Z < z) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Percentile</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mstyle></math>. Use the Inverse normal to find that <font size="3" face="Times New Roman"><em>z</em> = $CorrectZ</font> . Since this is not <font size="3" face="Times New Roman">$WrongZ</font>, the statement is false.</p>
<p>&nbsp;</p>@
qu.1.18.editing=useHTML@
qu.1.18.solution=@
qu.1.18.algorithm=$Q=12;
$Percentile = range(51,60,1);
$PDigit=$Percentile-10*int($Percentile/10);
$Ending=switch($PDigit,"th","st","nd","rd","th","th","th","th","th","th");
$PreZ=maple("(stats[statevalf, icdf, normald])(1-$Percentile/100)");
$CorrectZ=decimal(4,$PreZ);
$WrongZ=decimal(4,(switch(rint(2),0,0.6)+range(0.5,0.9))*$CorrectZ);@
qu.1.18.uid=be1affcd-6b85-4d48-b89b-c3e13152c9e5@
qu.1.18.info=  Course=202;
  Type=T/F;
@
qu.1.18.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>The z value that corresponds to the $Percentile$Ending percentile is z = $WrongZ .</div>@
qu.1.18.answer=2@
qu.1.18.choice.1=True@
qu.1.18.choice.2=False@
qu.1.18.fixed=@

qu.1.19.mode=Inline@
qu.1.19.name=29. Binomial viz Normal, < vs ≤@
qu.1.19.comment=<p>This comes from the use of the continuity correction when approximation a binomial with a normal distribution. In fact:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&asymp;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$L1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$L2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn></mrow></mfenced></mrow></mstyle></math> while<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&asymp;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$L1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$L2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn></mrow></mfenced></mrow></mstyle></math></p>@
qu.1.19.editing=useHTML@
qu.1.19.solution=@
qu.1.19.algorithm=$Q=29;
$n=range(350,450,5);
$p=range(0.35,0.65,0.1);
$L1=range(int($p*$n)-10,int($p*$n)-5);
$L2=range(int($p*$n)+10,int($p*$n)+10);@
qu.1.19.uid=f8a746a0-47a3-473c-8bd7-9be20c02fe65@
qu.1.19.info=  Course=202;
  Type=MC;
  Keyword=binomial approximation;
@
qu.1.19.weighting=1@
qu.1.19.numbering=alpha@
qu.1.19.part.1.name=sro_id_1@
qu.1.19.part.1.editing=useHTML@
qu.1.19.part.1.choice.5=P(B) may be smaller or larger than P(A)@
qu.1.19.part.1.fixed=@
qu.1.19.part.1.choice.4=P(B) = P(A)<br>@
qu.1.19.part.1.question=null@
qu.1.19.part.1.choice.3=P(B) < P(A)<br>@
qu.1.19.part.1.choice.2=P(B) > P(A)<br>@
qu.1.19.part.1.choice.1=P(B) = 0.4<br>@
qu.1.19.part.1.mode=Multiple Choice@
qu.1.19.part.1.display=vertical@
qu.1.19.part.1.answer=2@
qu.1.19.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q">Suppose X has a binomial distribution with <font size="3" face="Times New Roman"><em>n</em> = $n</font> and <font size="3" face="Times New Roman"><em>p </em>= $p</font>, What can you say about the approximate probability of the event :<br /><br />A = {$L1 < X < $L2}&nbsp; compared to the approximate probability of the event:<br /><br />B = {$L1<title></title><meta content="Microsoft FrontPage 5.0" name="GENERATOR" /><meta content="FrontPage.Editor.Document" name="ProgId" />&le; X<title></title><meta content="Microsoft FrontPage 5.0" name="GENERATOR" /><meta content="FrontPage.Editor.Document" name="ProgId" />&le; $L2} ?<br /><span>&nbsp;</span><1><span> </span></div>@

qu.1.20.mode=Multiple Choice@
qu.1.20.name=23b. P(Z > x), find x@
qu.1.20.comment=<p>P(X > x) = $P means 1 - P(X < x) = $P so P(X < x) = 1 - $P = $Z&nbsp;</p>
<p>Standardizing:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z</mi></mrow></mstyle></math> so using the Inverse Normal we have :</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$I</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$I</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$U</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.20.editing=useHTML@
qu.1.20.solution=@
qu.1.20.algorithm=$Q="23b";
$U = range(5,9,1);
$SD = range(2,3,0.1);
$P = range(0.7,0.8,0.0001);
$Z = 1-$P;
$PreI = maple("(stats[statevalf,icdf,normald])($Z)");
$I=decimal(4,$PreI);
$Ans = decimal(4,$U + $SD*$I);
$Alt1=decimal(4,range(1.1,1.9,0.01)*$Ans);
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.20.uid=0ee23acb-a6f2-4fc8-b1ff-a2c8ff03e312@
qu.1.20.info=  Course=202;
  Type=MC;
@
qu.1.20.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>If X is a normal random variable with mean $U and standard deviation $SD, then find  the value x such that <br />
P(Z > x) is equal to $P, as shown below. (Note: the diagram is not necessarily to scale.)
<p>&nbsp;</p>
<p><img width="294" height="196" title="Normal graph, 70% [IMG:Normal70PC.gif]" src="__BASE_URI__CPD/ND/Normal70PC.gif" alt="" /></p>
</div>@
qu.1.20.answer=1@
qu.1.20.choice.1=$Ans@
qu.1.20.choice.2=$Alt1@
qu.1.20.choice.3=$Alt2@
qu.1.20.choice.4=$Alt3@
qu.1.20.fixed=@

qu.1.21.mode=Blanks@
qu.1.21.name=18 Mean & SD for standard@
qu.1.21.comment=<p>The Standard Normal, N(0,1) has <strong>mean</strong> = 0 and standard <strong>deviation</strong> = 1 .</p>@
qu.1.21.editing=useHTML@
qu.1.21.hint.1=Enter the actual <i>numeric</i> values of the mean and SD.@
qu.1.21.solution=@
qu.1.21.algorithm=@
qu.1.21.uid=a381582b-f9d7-4508-92ff-218b59444562@
qu.1.21.info=  Course=230;
  Difficulty=1;
  Type=FITB;
  Algorithmic=no;
@
qu.1.21.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q18"> In a standardized normal distribution  <1> mean is μ = 0 and the  <2> deviation is σ = 1 . </div> @
qu.1.21.blank.1=the@
qu.1.21.blank.2=standard@
qu.1.21.extra=@
qu.1.21.format.input=text@

qu.1.22.mode=Multiple Choice@
qu.1.22.name=22b. Mean, P(Z > x)  SD?@
qu.1.22.comment=<p>P(X > $X) = 1 - P(X < $X) so P(X < $X) = 1 - $P = $Z . Standardizing:&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi>SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z</mi></mrow></mstyle></math>so using the Inverse Normal we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi>SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$I</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>SD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$I</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.1.22.editing=useHTML@
qu.1.22.solution=@
qu.1.22.algorithm=$Q="22b";
$U = range(1,5,1);
$P = range(0.1,0.5,0.0001);
$X = range(6*$U/5,2*$U,0.01);
$Z = 1-$P;
$PreI = maple("(stats[statevalf,icdf,normald])($Z)");
$I=decimal(4,$PreI);
$Ans = decimal(4,($X - $U)/$I);
$Alt1=decimal(4,range(1.1,1.9,0.01)*$Ans);
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.22.uid=12e1188c-a4b1-44c7-81c4-70044c1a9e71@
qu.1.22.info=  Course=202;
  Type=MC;
@
qu.1.22.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>If X is a normal random variable with mean $U, and if the probability that X is more than $X is $P, then what is the standard deviation of X?
<p>&nbsp;</p>
</div>@
qu.1.22.answer=1@
qu.1.22.choice.1=$Ans@
qu.1.22.choice.2=$Alt1@
qu.1.22.choice.3=$Alt2@
qu.1.22.choice.4=$Alt3@
qu.1.22.fixed=@

qu.1.23.mode=Inline@
qu.1.23.name=26a. Binomial vs. Normal@
qu.1.23.comment=<p>Use the Normal approximation to the binomial with a continuity correction.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$N</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$P</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$U</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$N</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$P</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$S</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&simeq;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Z1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z1</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>Notice that without the continuity correction we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&simeq;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Z1WO</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2WO</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2WO</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z1WO</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P2WO</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P1WO</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$AnsWO</mi></mrow></mstyle></math></p>@
qu.1.23.editing=useHTML@
qu.1.23.hint.1=If you do not see your answer in the list, make sure you (properly) used a continuity correction!@
qu.1.23.solution=@
qu.1.23.algorithm=$Q = "26a";
$P = range(0.3,0.35,0.001);
$N = range(180,185);
$X1 = range(50,53);
$X2 = range(65,67);
$U = $P*$N;
$V = $N*$P*(1-$P);
$S=decimal(4,sqrt($V));
$Z1=decimal(4,($X1-0.5-$U)/$S);
$Z2=decimal(4,($X2+0.5-$U)/$S);
$Z1WO=decimal(4,($X1-$U)/$S);
$Z2WO=decimal(4,($X2-$U)/$S);
$MR = maple("MP1:=(stats[statevalf,cdf,normald]($Z1)):
MP2:=(stats[statevalf,cdf,normald]($Z2)):
MP1WO:=(stats[statevalf,cdf,normald]($Z1WO)):
MP2WO:=(stats[statevalf,cdf,normald]($Z2WO)):
MP1,MP2,MP1WO,MP2WO;
");
$P1=switch(0,$MR);
$P2=switch(1,$MR);
$P1WO=switch(2,$MR);
$P2WO=switch(3,$MR);
$Ans = decimal(4,$P2-$P1);
$AnsWO=decimal(4,$P2WO-$P1WO);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+$Alt1));
$Alt4=decimal(4,0.5*($Ans+$Alt2));
$PER = 100*$P;@
qu.1.23.uid=7d0b8e26-fa0e-4fc7-8b42-6b7828b788ee@
qu.1.23.info=  Type=numeric;
  Course=202;
  Keyword=continuity correction;
@
qu.1.23.weighting=1,1@
qu.1.23.numbering=alpha@
qu.1.23.part.1.name=sro_id_2@
qu.1.23.part.1.answer.units=@
qu.1.23.part.1.numStyle=thousands scientific  arithmetic@
qu.1.23.part.1.editing=useHTML@
qu.1.23.part.1.showUnits=false@
qu.1.23.part.1.err=0.0010@
qu.1.23.part.1.question=(Unset)@
qu.1.23.part.1.mode=Numeric@
qu.1.23.part.1.grading=toler_abs@
qu.1.23.part.1.negStyle=minus@
qu.1.23.part.1.answer.num=$Ans@
qu.1.23.part.2.name=sro_id_3@
qu.1.23.part.2.answer.units=@
qu.1.23.part.2.numStyle=thousands scientific  arithmetic@
qu.1.23.part.2.editing=useHTML@
qu.1.23.part.2.showUnits=false@
qu.1.23.part.2.err=0.0010@
qu.1.23.part.2.question=(Unset)@
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qu.1.23.part.2.grading=toler_abs@
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qu.1.23.part.2.answer.num=$AnsWO@
qu.1.23.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q">If the binomial distribution has N = $N and p = $P, then what is the approximate probability of P($X1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></mstyle></math>X <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></mstyle></math>$X2) if you use the continuity correction? (4 decimals)&nbsp; <span>&nbsp;</span><1><span>&nbsp;</span><p><span>Answer again NOT using the continuity correction: <span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p></div>@

qu.1.24.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>If X is a normal random variable with mean $U, and if the probability that X is more than $X is $P, then what is the standard deviation of X (3 decimal accuracy)?</div>@
qu.1.24.answer.num=$Ans@
qu.1.24.answer.units=@
qu.1.24.showUnits=false@
qu.1.24.grading=toler_abs@
qu.1.24.err=0.01@
qu.1.24.negStyle=minus@
qu.1.24.numStyle=thousands scientific dollars arithmetic@
qu.1.24.mode=Numeric@
qu.1.24.name=22a. Mean, P(Z > x)  SD?@
qu.1.24.comment=<p>P(X > $X) = 1 - P(X < $X) so P(X < $X) = 1 - $P = $Z . Standardizing:&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi>SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z</mi></mrow></mstyle></math>so using the Inverse Normal we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi>SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$I</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>SD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$I</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.1.24.editing=useHTML@
qu.1.24.solution=@
qu.1.24.algorithm=$Q="22a";
$U = range(1,5,1);
$P = range(0.1,0.5,0.0001);
$X = range(6*$U/5,2*$U,0.01);
$Z = 1-$P;
$PreI = maple("(stats[statevalf,icdf,normald])($Z)");
$I=decimal(4,$PreI);
$Ans = decimal(4,($X - $U)/$I);@
qu.1.24.uid=fef8fbfc-a9c2-4222-a6f0-7bda27f66dc5@
qu.1.24.info=  Course=202;
  Type=numeric;
@

qu.1.25.mode=Multiple Choice@
qu.1.25.name=26b. Binomial vs. Normal@
qu.1.25.comment=<p>Use the Normal approximation to the binomial with a continuity correction.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>n</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$N</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$P</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$U</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$N</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$P</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$S</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&simeq;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Z1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z1</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>Notice that without the continuity correction we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&simeq;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Z1WO</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2WO</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2WO</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z1WO</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P2WO</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P1WO</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$AnsWO</mi></mrow></mstyle></math></p>@
qu.1.25.editing=useHTML@
qu.1.25.hint.1=If you do not see your answer in the list, make sure you (properly) used a continuity correction!@
qu.1.25.solution=@
qu.1.25.algorithm=$Q = "26b";
$P = range(0.3,0.35,0.001);
$N = range(180,185);
$X1 = range(50,53);
$X2 = range(65,67);
$U = $P*$N;
$V = $N*$P*(1-$P);
$S=decimal(4,sqrt($V));
$Z1=decimal(4,($X1-0.5-$U)/$S);
$Z2=decimal(4,($X2+0.5-$U)/$S);
$Z1WO=decimal(4,($X1-$U)/$S);
$Z2WO=decimal(4,($X2-$U)/$S);
$MR = maple("MP1:=(stats[statevalf,cdf,normald]($Z1)):
MP2:=(stats[statevalf,cdf,normald]($Z2)):
MP1WO:=(stats[statevalf,cdf,normald]($Z1WO)):
MP2WO:=(stats[statevalf,cdf,normald]($Z2WO)):
MP1,MP2,MP1WO,MP2WO;
");
$P1=switch(0,$MR);
$P2=switch(1,$MR);
$P1WO=switch(2,$MR);
$P2WO=switch(3,$MR);
$Ans = decimal(4,$P2-$P1);
$AnsWO=decimal(4,$P2WO-$P1WO);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+$Alt1));
$Alt4=decimal(4,0.5*($Ans+$Alt2));
$PER = 100*$P;@
qu.1.25.uid=0329bd09-0fa4-4305-834a-b795c4482582@
qu.1.25.info=  Type=MC;
  Course=202;
  Keyword=continuity correction;
@
qu.1.25.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q">If the binomial distribution has N = $N and p = $P, then what is the approximate probability of P($X1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></mstyle></math>X <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></mstyle></math>$X2)?
<p>&nbsp;</p>
</div>@
qu.1.25.answer=1@
qu.1.25.choice.1=$Ans@
qu.1.25.choice.2=$Alt1@
qu.1.25.choice.3=$Alt2@
qu.1.25.choice.4=$Alt3@
qu.1.25.choice.5=$Alt4@
qu.1.25.fixed=@

qu.1.26.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution/Q$Q"><a onclick="window.open(this.href,this.target,'height=150,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>Let Z ~ N(0,1). Find Pr($z1 &le; Z &le; $z2) (2 decimal accuracy please.)<br />
<br />
<strong>NOTE:</strong> Maple TA is intelligent enough to accept and evaluate arithmetic expressions. For example <strong>0.3</strong> and <strong>1 - 0.7</strong> would both be considered as the same answer.</div>@
qu.1.26.answer.num=$Ans@
qu.1.26.answer.units=@
qu.1.26.showUnits=false@
qu.1.26.grading=toler_abs@
qu.1.26.err=0.01@
qu.1.26.negStyle=minus@
qu.1.26.numStyle=thousands scientific dollars arithmetic@
qu.1.26.mode=Numeric@
qu.1.26.name=04. Find P(a < Z < b) algorithmic@
qu.1.26.comment=<p>P($z1 < z < $z2) = P(z < $z2) - P(z < $z1) = $Pz2 - $Pz1 = $Ans</p>@
qu.1.26.editing=useHTML@
qu.1.26.solution=@
qu.1.26.algorithm=$Q=4;
$z1=decimal(2,range(-2,1.3,0.01));
$z2=decimal(2,range($z1+.05,1.6,0.01));
$Pz=maple("A:=(stats[statevalf, cdf, normald])($z1):
B:=(stats[statevalf, cdf, normald])($z2):
[A,B]");
$Pz1=decimal(3,switch(0,$Pz));
$Pz2=decimal(3,switch(1,$Pz));
$Ans=$Pz2 - $Pz1;@
qu.1.26.uid=ea08cae0-996f-4ae7-9e11-300d298fd28b@
qu.1.26.info=  Course=230;
@

qu.1.27.mode=True False@
qu.1.27.name=05. Area of Z > z@
qu.1.27.comment=<p>First use a table or calculator to determine that <font size="3" face="Times New Roman"><em>F</em>($CutOff) = $FC </font>.</p>
<p>The area to the right of <font size="3" face="Times New Roman">$CutOff</font> then is <font size="3" face="Times New Roman">1 - $FC = $Ans</font> which of course is not the same as <font size="3" face="Times New Roman">$NGAns</font>.</p>@
qu.1.27.editing=useHTML@
qu.1.27.solution=@
qu.1.27.algorithm=$Q=5;
$CutOff=range(1.5,1.92,0.01);
$PreFC=maple("(stats[statevalf, cdf, normald])($CutOff)");
$FC=decimal(4,$PreFC);
$Ans=1-$FC;
$NGAns=decimal(4,$Ans*range(0.35,0.75,0.01));@
qu.1.27.uid=87469459-4b21-4c07-9694-d55844f5bc76@
qu.1.27.info=  Course=230;
  Course=202;
  Type=TF;
@
qu.1.27.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>The area to the right of <font size="3" face="Times New Roman"><em>z</em> = $CutOff </font>is <font size="3" face="Times New Roman">$NGAns</font>. (<em><font size="3" face="Times New Roman">Z</font></em> has the Standard Normal distribution.)</div>@
qu.1.27.answer=2@
qu.1.27.choice.1=True@
qu.1.27.choice.2=False@
qu.1.27.fixed=@

qu.1.28.mode=Multiple Choice@
qu.1.28.name=10. μ, σ, P(Z < x) : what is x?@
qu.1.28.comment=<p>First, find the <em>Inverse Normal</em> value for $p - that is, what value of <em><font size="3" face="Times New Roman">z</font></em> has <font size="3" face="Times New Roman"><em>P</em>(<em>Z</em> < <em>z</em>) = $P</font> where <font size="3" face="Times New Roman"><em>Z</em></font> is the standard normal. The answer is $ZSpot. Now, standardizing the given data we get:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$ZSpot</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Mean</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$SD</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$ZSpot</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.28.editing=useHTML@
qu.1.28.solution=@
qu.1.28.algorithm=$Q=10;
$Mean=range(4,9,0.25);
$SD=range(1,$Mean/2,0.1);
$p=range(0.81,0.87,0.05);
$Z = maple("(stats[statevalf, icdf, normald])($p)");
$ZSpot=decimal(4,$Z);
$Ans=decimal(4,$Mean+$SD*$ZSpot);
$Alt1=decimal(4,$Ans+$SD+range(0.0001,0.003,0.0001));
$Alt2=decimal(4,$Ans-$SD-range(0.0001,0.003,0.0001));
$Alt3=0.5*($Ans+switch(rint(2),$Alt1,$Alt2));@
qu.1.28.uid=2ae366e1-36ea-4066-97bd-2b5ebff74507@
qu.1.28.info=  Course=202;
  Type=MC;
@
qu.1.28.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>If X is a normal random variable with mean $Mean and standard deviation $SD, then find  the value x such that P(Z < x) is equal to $p, as shown below. (Note: the diagram is not necessarily to scale.)<br />
<p><img width="289" height="189" src="__BASE_URI__CPD/ND/NormalLeft85PC.gif" alt="Normal curve, 85% point." title="Normal curve, 85% point. [IMG:NormalLeft85PC.gif]" /></p>
</div>@
qu.1.28.answer=1@
qu.1.28.choice.1=$Ans@
qu.1.28.choice.2=$Alt1@
qu.1.28.choice.3=$Alt2@
qu.1.28.choice.4=$Alt3@
qu.1.28.fixed=@

qu.1.29.mode=Multiple Choice@
qu.1.29.name=02. P(Z > z)@
qu.1.29.comment=<p><font size="3" face="Times New Roman"><em>P</em>(<em>z</em> > -$Z) = 1 - <em>P</em>(<em>z</em> < -$Z) = 1 - $PZ = $Ans</font></p>@
qu.1.29.editing=useHTML@
qu.1.29.solution=@
qu.1.29.algorithm=$Q=2;
$Z = decimal(2,range(0.1,0.9,0.01));
$PrePZ = maple("(stats[statevalf,cdf,normald])(-$Z)");
$PZ = decimal(4,$PrePZ);
$Ans=1-$PZ;
$Alt1 = decimal(4,$Ans+(1-$Ans)*range(0.4,0.8,0.01));
$Alt2 = decimal(4,$Ans*range(0.4,0.9,0.01));
$Alt3 = decimal(4,(switch(rint(2),$Alt1,$Alt2)+$Ans)/2);@
qu.1.29.uid=693b06d0-d066-4fd8-a2ff-7414e0d077b4@
qu.1.29.info=  Course=230;
  Course=202;
  Type=MC;
@
qu.1.29.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>Let <font size="3" face="Times New Roman"><em>Z</em> ~ <em>N</em>(0,1)</font>. Then&nbsp; <font size="3" face="Times New Roman"><em>P</em>(<em>z</em> > &ndash;$Z)</font> is:</div>@
qu.1.29.answer=1@
qu.1.29.choice.1=$Ans@
qu.1.29.choice.2=$Alt1@
qu.1.29.choice.3=$Alt2@
qu.1.29.choice.4=$Alt3@
qu.1.29.fixed=@

qu.1.30.mode=Multiple Choice@
qu.1.30.name=20. Property of two + z scores?@
qu.1.30.comment=Remember that the standard normal curve has half its area, that is 0.5, to the right of zero. So any two points from the right of zero must have half or less of the curve's area between them.<br />
<img align="center" src="__BASE_URI__CPD/ND/NormalCurve3.gif" alt="Normal Curve" title="Normal Curve [IMG:NormalCurve3.gif]" /></p>@
qu.1.30.editing=useHTML@
qu.1.30.solution=@
qu.1.30.algorithm=$Q=20;
$Fake=range(20,33,1);@
qu.1.30.uid=cb5c4876-9835-4ae1-9317-b79599733ec3@
qu.1.30.info=  Type=MC;
  Course=230;
@
qu.1.30.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q">Given any two positive standardized z-scores of a standard normal distribution, which of the following statements must be true?</div>@
qu.1.30.answer=1@
qu.1.30.choice.1=The percent of data between these scores is less than 50.@
qu.1.30.choice.2=The percent of data greater than the larger score is more than $Fake@
qu.1.30.choice.3=The percent of data less than the smaller score is less than $Fake.@
qu.1.30.choice.4=The percent of data between the scores is more that $Fake@
qu.1.30.choice.5=The percent of data less than the smaller score equals the percent of data greater than the larger score.@
qu.1.30.fixed=@

qu.1.31.mode=Multiple Choice@
qu.1.31.name=15b. Given Area(μ - z) = x%,  z is?@
qu.1.31.comment=<p>You know that (with the standard Normal) P(Z &le; 0) = 0.5 .&nbsp; There are <u>two</u> possible z values with $Percent% of the data lying between them and the mean, however they are just negatives of each other so we can just find one of them.</p>
<p>&nbsp;</p>
<p>Consider z with P(Z &le; z) = 0.5 + $Percent&nbsp; . Use the Inverse Normal to find z = $Ans. Notice that z = -$Ans is also a valid answer.</p>@
qu.1.31.editing=useHTML@
qu.1.31.solution=@
qu.1.31.algorithm=$Q="15b";
$Percent = range(15,20,1);
$P = $Percent/100;
$Z = 0.5 + $P;
$I = maple("(stats[statevalf,icdf,normald])($Z)");
$Ans = decimal(4,$I);
$Alt1=decimal(4,range(1.1,1.9,0.01)*$Ans);
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.31.uid=eaf00155-a1c4-49cb-a92f-979367c03f7d@
qu.1.31.info=  Course=202;
  Type=MC;
@
qu.1.31.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>Consider the standard normal Z ~ N(0,1). What z&nbsp; value has $Percent% of the data lying between itself and the mean?</div>@
qu.1.31.answer=1@
qu.1.31.choice.1=$Ans@
qu.1.31.choice.2=$Alt1@
qu.1.31.choice.3=$Alt2@
qu.1.31.choice.4=$Alt3@
qu.1.31.fixed=@

qu.1.32.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a> Find the z value that corresponds to the $Percentile$Ending percentile (3 decimals). <br />
<p>(Z is the standard normal)</p>
</div>@
qu.1.32.answer.num=$CorrectZ@
qu.1.32.answer.units=@
qu.1.32.showUnits=false@
qu.1.32.grading=toler_abs@
qu.1.32.err=.01@
qu.1.32.negStyle=minus@
qu.1.32.numStyle=thousands scientific dollars arithmetic@
qu.1.32.mode=Numeric@
qu.1.32.name=14a. Find the nth percentile@
qu.1.32.comment=<p>"z corresponding to the $Percentile$Ending percentile" just means that P(Z < z) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Percentile</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mstyle></math>. Use the Inverse normal to find that <font size="3" face="Times New Roman"><em>z</em> = $CorrectZ</font> .</p>
<p>&nbsp;</p>@
qu.1.32.editing=useHTML@
qu.1.32.solution=@
qu.1.32.algorithm=$Q="14a";
$Percentile = range(40,60,1);
$PDigit=$Percentile-10*int($Percentile/10);
$Ending=switch($PDigit,"th","st","nd","rd","th","th","th","th","th","th");
$PreZ=maple("(stats[statevalf, icdf, normald])($Percentile/100)");
$CorrectZ=decimal(4,$PreZ);@
qu.1.32.uid=1667c219-d423-439e-aab5-ef53a4af0480@
qu.1.32.info=  Course=202;
  Type=numeric;
  Keyword=percentile;
@

qu.1.33.mode=Multiple Choice@
qu.1.33.name=17b. Mean of Sample@
qu.1.33.comment=<p>The expected value of the mean of samples of any size is just the distribution's mean.</p>@
qu.1.33.editing=useHTML@
qu.1.33.solution=@
qu.1.33.algorithm=$Q="17b";
$SS=range(5,12,1);
$Mean=range(20,50,1);
$SD=range(3,$Mean/2,1);
$Ans=$Mean;
$Alt1=int(range(1.1,1.9,0.01)*$Ans);
$Alt2=int(range(0.5,0.9,0.01)*$Ans);
$Alt3=int(0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.33.uid=9b37a7ed-1328-4727-88a7-1b1dc72f4e8d@
qu.1.33.info=  Course=202;
  Type=MC;
@
qu.1.33.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q">A normal population (represented by the r.v. X) has mean $Mean and standard deviation $SD. For a random sample of size $SS, determine the mean of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mover accent='false'><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>X</mi></mrow><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo></mrow></mover></mrow></mrow></math>.</div>@
qu.1.33.answer=1@
qu.1.33.choice.1=$Ans@
qu.1.33.choice.2=$Alt1@
qu.1.33.choice.3=$Alt2@
qu.1.33.choice.4=$Alt3@
qu.1.33.fixed=@

qu.1.34.mode=Multiple Choice@
qu.1.34.name=19. Area under curve@
qu.1.34.comment=<p>P(-$Z < z < $Z) = P(z < $Z) - P(z < -$Z) = $PZ - $PMZ = $Ans</p>@
qu.1.34.editing=useHTML@
qu.1.34.solution=@
qu.1.34.algorithm=$Q=23;
$Z=range(1.75,2.25,0.05);
$L=maple("[(stats[statevalf, cdf, normald])($Z),
(stats[statevalf, cdf, normald])(-$Z)]");
$PZ=decimal(4,switch(0,$L));
$PMZ=decimal(4,switch(1,$L));
$Ans=decimal(4,$PZ-$PMZ);
$Alt1=decimal(4,range($Ans+0.01,0.99,0.001));
$Alt2=decimal(4,range(0.15,$Ans-0.05,0.001));
$Which=switch(rint(2),$Alt1,$Alt2);
$Alt3=decimal(4,($Ans+$Which)/2);@
qu.1.34.uid=e561b8c2-f0ef-4402-aae5-017d2d784e60@
qu.1.34.info=  Course=202;
  Type=MC;
@
qu.1.34.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q">Find the area under the curve between z = &minus;$Z and z = $Z.
<p><applet width="218" height="127" code="applets.labelImage.LabelImage">
<param value="__BASE_URI__CPD/ND/NormalCurve.gif" name="image" />
<param value="2" name="size" />
<param value="45" name="label.1.x" />
<param value="118" name="label.1.y" />
<param value="-$Z" name="label.1.text" />
<param value="169" name="label.2.x" />
<param value="118" name="label.2.y" />
<param value="$Z" name="label.2.text" /> </applet></p>
<a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a></div>@
qu.1.34.answer=1@
qu.1.34.choice.1=$Ans@
qu.1.34.choice.2=$Alt1@
qu.1.34.choice.3=$Alt2@
qu.1.34.choice.4=$Alt3@
qu.1.34.fixed=@

qu.1.35.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Mathematic/Q$Q">A random sample of&nbsp;$N observations is to be drawn from a population with a mean of&nbsp;$U and a standard deviation of $S. The probability that the mean of the sample will exceed&nbsp;$X is (4 decimal accuracy):</div>@
qu.1.35.answer.num=$Ans@
qu.1.35.answer.units=@
qu.1.35.showUnits=false@
qu.1.35.grading=toler_abs@
qu.1.35.err=.001@
qu.1.35.negStyle=minus@
qu.1.35.numStyle=thousands scientific dollars arithmetic@
qu.1.35.mode=Numeric@
qu.1.35.name=28a. P(Mean > z)@
qu.1.35.comment=<p>Let the mean of the sample be X. Then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.35.editing=useHTML@
qu.1.35.solution=@
qu.1.35.algorithm=$Q="28a";
$N = range(100,110);
$U = range(40,45);
$S = range(25,30);
$X = range($U+1,$U+3);
$Z = decimal(4,($X-$U)/($S/sqrt($N)));
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$Ans = decimal(4,1-$P);@
qu.1.35.uid=e0399119-3454-457b-a9ac-476ee9c4ac14@
qu.1.35.info=  Course=202;
  Type=numeric;
@

qu.1.36.mode=Multiple Choice@
qu.1.36.name=14b. Find nth percentile@
qu.1.36.comment=<p>"z corresponding to the $Percentile$Ending percentile" just means that P(Z < z) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Percentile</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mstyle></math>. Use the Inverse normal to find that <font size="3" face="Times New Roman"><em>z </em>= $Ans</font> .</p>
<p>&nbsp;</p>@
qu.1.36.editing=useHTML@
qu.1.36.solution=@
qu.1.36.algorithm=$Q="14b";
$Percentile = range(40,60,1);
$PDigit=$Percentile-10*int($Percentile/10);
$Ending=switch($PDigit,"th","st","nd","rd","th","th","th","th","th","th");
$PreZ=maple("(stats[statevalf, icdf, normald])(1-$Percentile/100)");
$Ans=decimal(4,$PreZ);
$Alt1=decimal(4,range(0.5,0.9)*$Ans);
$Alt2=decimal(4,range(1.1,1.5)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.36.uid=2e3f6ab1-4186-408f-889b-50e3876e71bd@
qu.1.36.info=  Course=202;
  Type=MC;
  Keyword=percentile;
@
qu.1.36.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>Find the z value that corresponds to the $Percentile$Ending percentile (3 decimals). <br />
<p>(Z is the standard normal)</p>
</div>@
qu.1.36.answer=1@
qu.1.36.choice.1=$Ans@
qu.1.36.choice.2=$Alt1@
qu.1.36.choice.3=$Alt2@
qu.1.36.choice.4=$Alt3@
qu.1.36.fixed=@

qu.1.37.mode=True False@
qu.1.37.name=13. z the nth percentile ? (True)@
qu.1.37.comment=<p>"z corresponding to the $Percentile$Ending percentile" just means that P(Z < z) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$Percentile</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mstyle></math>. Use the Inverse normal to find that <font size="3" face="Times New Roman">z = $CorrectZ</font> . Since this is what the question says, the statement is true.</p>
<p>&nbsp;</p>@
qu.1.37.editing=useHTML@
qu.1.37.solution=@
qu.1.37.algorithm=$Q=13;
$Percentile = range(40,60,1);
$PDigit=$Percentile-10*int($Percentile/10);
$Ending=switch($PDigit,"th","st","nd","rd","th","th","th","th","th","th");
$PreZ=maple("(stats[statevalf, icdf, normald])(1-$Percentile/100)");
$CorrectZ=-decimal(4,$PreZ);
$WrongZ=decimal(4,(switch(rint(2),0,0.6)+range(0.5,0.9))*$CorrectZ);@
qu.1.37.uid=e91cf80e-ca35-499d-bcf5-0c42f8bdca49@
qu.1.37.info=  Course=202;
  Type=T/F;
@
qu.1.37.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Mathematic/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>The z value that corresponds to the $Percentile$Ending percentile is z = $CorrectZ .</div>@
qu.1.37.answer=1@
qu.1.37.choice.1=True@
qu.1.37.choice.2=False@
qu.1.37.fixed=@

qu.2.topic=Central Limit Theorem@

qu.2.1.question=<div title="UW Statistics Bank/Continuous Probability Models/Central Limit Theorem/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img hspace="4" border="0" align="$CalcAlign" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>The unemployment rate in a city is $PD%. A sample of $N persons is selected from the labor force. Find (3 decimals) the probability that less than $EX unemployed people are in the sample.</div>@
qu.2.1.answer.num=$Ans@
qu.2.1.answer.units=@
qu.2.1.showUnits=false@
qu.2.1.grading=toler_abs@
qu.2.1.err=0.01@
qu.2.1.negStyle=minus@
qu.2.1.numStyle=thousands scientific dollars arithmetic@
qu.2.1.mode=Numeric@
qu.2.1.name=05. P(# unemployed < n)@
qu.2.1.comment=<p><strong>The correct answer is $Ans .</strong></p>
<p>Let X be the number of unemployed in the sample. Treat this as a binomial distribution so X ~ Bin($N,$P). Use the Normal Approximation to the Binomial with p = $P and n = $N . The Standard Normal approximation then is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>W</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$N</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$P</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$P</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Q</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$NP</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$NPQ</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math> so <strong>applying the Continuity Correction </strong>we get:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$EX</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$EXM1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&asymp;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>W</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$EXM1CC</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$NP</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$NPQ</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>W</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$W</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p><strong>NOTE:</strong> If your answer was $AnswoCC, then you failed to apply the Continuity Correction (that is you used $EXM1 instead of $EXM1CC in the expression above).</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$Q=5;
$P=decimal(3,range(0.025,0.1,.001));
$PD=100*$P;
$N=range(200,800,25);
$NP=$N*$P;
$EX=range(int(4*$NP/5),int(6*$NP/5),5);
$EXM1=$EX - 1;
$EXM1CC=$EXM1+0.5;
$NPQ=$NP*(1-$P);
$W=($EXM1CC-$NP)/sqrt($NPQ);
$WwoCC=($EXM1-$NP)/sqrt($NPQ);
$PreAns=maple("(stats[statevalf, cdf, normald])($W)");
$PreAnswoCC=maple("(stats[statevalf, cdf, normald])($WwoCC)");
$Ans=decimal(3,$PreAns);
$AnswoCC=decimal(3,$PreAnswoCC);
$CalcAlign=switch(rint(2),"Left","Right");@
qu.2.1.uid=5d1c5921-2970-44c2-9467-2342b4595a43@
qu.2.1.info=  Course=230;
  Keyword=continuity correction;
  Type=numeric;
@

qu.2.2.question=<div title="UW Statistics Bank/Continuous Distributions/Central Limit Theorem/Q$Q">
A wholesale distributor has found that the amount of a customer's order is a normal random variable with a mean of \\$$U&nbsp;and a standard deviation of \\$$S. What is the probability that the total amount in a random sample of $N orders is greater than \\$$X? (4 decimal accuracy)</div>@
qu.2.2.answer.num=$Ans@
qu.2.2.answer.units=@
qu.2.2.showUnits=false@
qu.2.2.grading=toler_abs@
qu.2.2.err=.001@
qu.2.2.negStyle=minus@
qu.2.2.numStyle=thousands scientific dollars arithmetic@
qu.2.2.mode=Numeric@
qu.2.2.name=08. Total Order $>z@
qu.2.2.comment=<p>This is a Central Limit Theorem problem. Our mean is n&mu; = $N($U) = $NU, the SD is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>$S</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SD</mi></mrow></mstyle></math>. Let <font size="3" face="Times New Roman"><em>X<sub>i</sub></em></font> represent the value of the individual orders.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$N</mi></mrow></munderover><msub><mi>X</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$N</mi></mrow></munderover><msub><mi>X</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$NU</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$NU</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$PD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$Q=8;
$U = range(200,205);
$S = range(55,60);
$N = range(20,22);
$NU=$N*$U;
$SD=decimal(4,sqrt($N*$S^2));
$X = range(4511,4520);
$Z = ($X-$U*$N)/($S*sqrt($N));
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$PD=decimal(4,$P);
$Ans = decimal(4,1-$P);@
qu.2.2.uid=25d39a25-65b1-46e8-b5f4-07989bd4935c@
qu.2.2.info=  Course=202;
  Type=numeric;
@

qu.2.3.mode=Inline@
qu.2.3.name=06. Conditions for CLT@
qu.2.3.comment=@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$Q=6;
$Sheets=range(100,250,10);
$Alt3=switch(rint(3),"the law of proportions","Mazlos Hierarchy","convergence of least-likely events");@
qu.2.3.uid=3225e2a9-0e35-4ef7-90f9-883c08091938@
qu.2.3.info=  Course=230;
@
qu.2.3.weighting=1@
qu.2.3.numbering=alpha@
qu.2.3.part.1.comment.3=@
qu.2.3.part.1.comment.2=@
qu.2.3.part.1.name=sro_id_1@
qu.2.3.part.1.comment.1=@
qu.2.3.part.1.editing=useHTML@
qu.2.3.part.1.choice.5=none of the above.<br>@
qu.2.3.part.1.fixed=4@
qu.2.3.part.1.choice.4=the fact that probability is the long run proportion of times an event occurs@
qu.2.3.part.1.question=null@
qu.2.3.part.1.choice.3=$Alt3@
qu.2.3.part.1.choice.2=the central limit theorem@
qu.2.3.part.1.choice.1=the law of large numbers@
qu.2.3.part.1.mode=Multiple Choice@
qu.2.3.part.1.display=vertical@
qu.2.3.part.1.comment.5=@
qu.2.3.part.1.comment.4=@
qu.2.3.part.1.answer=2@
qu.2.3.question=<div title="UW Statistics Bank/Continuous Probability Models/Central Limit Theorem/Q$Q">A factory produces plate glass with a mean thickness of 4 mm and a standard deviation of 1.1 mm. A simple random sample of $Sheets sheets of glass is to be measured, and the sample mean thickness of the $Sheets sheets is to be computed. We know the random variable has approximately a normal distribution because of what? <br /><p><span> </span><1><span> </span></p></div>@

qu.2.4.question=<div title="UW Statistics Bank/Continuous Probability Models/Central Limit Theorem/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.html" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a> Suppose X is a continuous random variable with probability density function:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>5</mn></mrow></mfrac></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&amp;comma;</mo><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msqrt><mrow><mn>5</mn></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mstyle></math><br />
Suppose we have $SS independent observations from this distribution. Find the approximate probability that their average (sample mean) lies between 0.04 and 0.06. 3 decimal accuracy please.</div>@
qu.2.4.answer.num=$ans@
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qu.2.4.grading=toler_abs@
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qu.2.4.negStyle=minus@
qu.2.4.numStyle=thousands scientific dollars arithmetic@
qu.2.4.mode=Numeric@
qu.2.4.name=04. Estimate mean@
qu.2.4.comment=<p>This question is best done using the <em>Central Limit Theorem</em>. To do that you need to find Var(Y), which in turn requires E(Y) and E(Y<sup>2</sup>):</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>Y</mi><mrow></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></munderover><mi>y</mi><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>5</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi mathcolor='#800080'>x</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn><mo lspace='0.0em' rspace='0.0em'></mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mfrac><msubsup><mfenced open='(' close=')' separators=','><mrow><mfrac><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><msup><mi>y</mi><mrow><mn>4</mn></mrow></msup><mrow><mn>20</mn></mrow></mfrac></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></msubsup></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mfrac><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>5</mn><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mfrac><mn>25</mn><mrow><mn>20</mn></mrow></mfrac></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>5</mn><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><mn>25</mn><mrow><mn>20</mn></mrow></mfrac><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>Y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></munderover><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>5</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi mathcolor='#800080'>x</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
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<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mfrac><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mn>5</mn><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>5</mn><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><msqrt><mrow><mn>5</mn></mrow></msqrt><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mfenced></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>5</mn><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mfrac><mn>5</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#8901;</mo><mfrac><mn>4</mn><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math></p>
<p>So Var(Y) = 1 - 0 = 1.<br />
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<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$LeftL</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>0</mn></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$VarIs</mi></mrow></msqrt></mrow></mfrac><mrow><mo lspace='0.0em' rspace='0.0em'>&#8804;</mo><mfrac><mrow><mfrac><mn>1</mn><mi mathvariant='normal'>$SS</mi></mfrac><mrow><munderover><mi>&Sum;</mi><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$SS</mi></mrow></munderover><msub><mi mathvariant='normal'>$Y</mi><mi>i</mi></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>0</mn></mrow></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$VarIs</mi></mrow></msqrt></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#8804;</mo><mfrac><mrow><mi mathvariant='normal'>$RightL</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>0</mn></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$VarIs</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mfenced></mrow></mrow></mstyle></math><br />
=P($SLeftL &le; Z &le; $SRightL) where "Z" is standard Normal.<br />
= F($SRightL) - F($SLeftL) Use a table, or the calculator provided <br />
= $FRight - $FLeft = $ans</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$Q=4;
$SS=range(80,200,10);
$VarIs=decimal(4,1/$SS);
$SDIs=decimal(4,sqrt($VarIs));
$LeftL=0.04;
$RightL=0.06;
$SLeftL=decimal(3,$LeftL/$SDIs);
$SRightL=decimal(3,$RightL/$SDIs);
$FRight=maple("(stats[statevalf, cdf, normald])($SRightL)");
$FR=decimal(3,$FRight);
$FLeft=maple("(stats[statevalf, cdf, normald])($SLeftL)");
$LR=decimal(3,$FLeft);
$ans=decimal(3,$FRight-$FLeft);@
qu.2.4.uid=57b5d8c3-acc9-4bcf-8321-80d7650cecb0@
qu.2.4.info=  Course=230;
  Type=numeric;
@

qu.2.5.mode=Multiple Choice@
qu.2.5.name=09. Ave. Passenger Weight@
qu.2.5.comment=<p>This is a Central Limit Theorem problem. Let X<sub>i</sub> represent the passenger weights. We will standardize to a Normal distribution with: <br />
<br />
<font size="3" face="Times New Roman">Mean = $U($N) = $U1 and&nbsp; SD = $SD*$N = $S1</font> so:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$N</mi></mrow></munderover><msub><mi>X</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$N</mi></mrow></munderover><msub><mi>X</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$N</mi></mrow></munderover><msub><mi>X</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U1</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U1</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=$Q=9;
$U = range(76,81);
$SD = range(15,25);
$N = range(90,100);
$U1 = $U*$N;
$S1 = $SD*sqrt($N);
$X = range(7900,8100,5);
$Z = decimal(4,($X-$U1)/($S1));
$PreP = maple("(stats[statevalf,cdf,normald])($Z)");
$P=decimal(4,$PreP);
$Ans = decimal(4,1-$P);
condition:gt($Ans,0.01);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+$Alt1));
$Alt4=decimal(4,0.5*($Ans+$Alt2));@
qu.2.5.uid=3c0ef1b7-8924-42f5-aa90-0d3f0480f323@
qu.2.5.info=  Type=MC;
  Course=230;
@
qu.2.5.question=<div title="UW Statistics Bank/Continuous Distributions/Central Limit Theorem/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:calculator.gif]" /></a>An airplane is only allowed a gross passenger weight of&nbsp;$X kg. If the weights of passengers traveling by air between Toronto and Vancouver have a mean of&nbsp;$U kg and a standard deviation of&nbsp;$SD kg, the approximate probability that the combined weight of $N passengers will exceed&nbsp;$X kg is:</div>@
qu.2.5.answer=1@
qu.2.5.choice.1=$Ans@
qu.2.5.choice.2=$Alt1@
qu.2.5.choice.3=$Alt2@
qu.2.5.choice.4=$Alt3@
qu.2.5.choice.5=$Alt4@
qu.2.5.fixed=@

qu.2.6.question=<div title="UW Statistics Bank/Continuous Probability Models/Central Limit Theorem/Q$Q">An extremely large population has a mean of $Mean and a standard deviation of $SD. Consider all possible samples of size $SampleSize. What would be the value of the standard deviation of the sample means? (Please answer to 3 decimals of accuracy.)</div>@
qu.2.6.answer.num=$Ans@
qu.2.6.answer.units=@
qu.2.6.showUnits=false@
qu.2.6.grading=toler_abs@
qu.2.6.err=.001@
qu.2.6.negStyle=minus@
qu.2.6.numStyle=thousands scientific dollars arithmetic@
qu.2.6.mode=Numeric@
qu.2.6.name=01. Estimate SD of Sample Means@
qu.2.6.comment=<p>Let X be the r.v. representing the number of students infected. The <strong>Central Limit Theorem</strong> tells us that the Standard Deviation of the sample means is approximately:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>Population</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>SD</mi></mrow></mfenced></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>Sample Size</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SampleSize</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math></p>
<p>Notice that the mean (of the population) has no bearing on our answer!</p>@
qu.2.6.editing=useHTML@
qu.2.6.solution=@
qu.2.6.algorithm=$Q=1;
$Mean=range(10,100,1);
$SampleSize=range(150,400,1);
$SD=range(1,12,1);
$Ans=decimal(3,$SD/sqrt($SampleSize));@
qu.2.6.uid=96885d3e-c7f1-4513-8774-52ae6f6e5351@
qu.2.6.info=  Course=230;
  Type=numeric;
@

qu.2.7.question=<div title="UW Statistics Bank/Continuous Probability Models/Central Limit Theorem/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>Suppose X is a continuous random variable with probability density function:&nbsp;
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mrow><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><mi>x</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>otherwise</mi></mrow></mfrac><mrow><mfrac linethickness='0'><mrow><mi></mi></mrow><mrow><mi></mi></mrow></mfrac></mrow></mrow></mrow></mstyle></math><br />
<br />
&nbsp; <br />
Suppose we have $SampleSize independent observations from this distribution. Find the approximate probability that their average (sample mean) lies between $LeftLimit and $RightLimit. 3 decimal accuracy please.</p>
</div>@
qu.2.7.answer.num=$Ans@
qu.2.7.answer.units=@
qu.2.7.showUnits=false@
qu.2.7.grading=toler_abs@
qu.2.7.err=0.01@
qu.2.7.negStyle=minus@
qu.2.7.numStyle=thousands scientific dollars arithmetic@
qu.2.7.mode=Numeric@
qu.2.7.name=03. P(Sample Mean in Range)@
qu.2.7.comment=<p>The correct answer is $Ans</p>
<p>This requires an application of the <em>Central Limit Theorem</em>. To do that you need Var(X), which in turn requires E(X) and E(X<sup>2</sup>).</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi>&#8734;</mi></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>&#8734;</mi></mrow></munderover><mi>x</mi><mi mathcolor='#0000ff'>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi mathcolor='#800080'>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></munderover><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi mathcolor='#800080'>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><msubsup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'> </mo><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><msup><mi>x</mi><mrow><mn>4</mn></mrow></msup><mrow><mn>4</mn></mrow></mfrac></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#8901;</mo><mfrac><mn>4</mn><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mi>&#8734;</mi></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&#8734;</mi></mrow></mrow></munderover><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mi mathcolor='#0000ff'>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi mathcolor='#800080'>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></munderover><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi mathcolor='#800080'>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><msubsup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'> </mo><msup><mi>x</mi><mrow><mn>4</mn></mrow></msup></mrow><mrow><mn>4</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><msup><mi>x</mi><mrow><mn>5</mn></mrow></msup><mrow><mn>5</mn></mrow></mfrac></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#8901;</mo><mfrac><mn>8</mn><mrow><mn>5</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>6</mn><mrow><mn>5</mn></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Var</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>X</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>5</mn></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p>To use the CLT we need:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Var</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$SampleSize</mi></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>&ImaginaryI;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$SampleSize</mi></mrow></munderover><msub><mi>X</mi><mrow><mi>&ImaginaryI;</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><msup><mi mathvariant='normal'>$SampleSize</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>&ImaginaryI;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>$SampleSize</mi></mrow></munderover><mi>Var</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>X</mi><mrow><mi>&ImaginaryI;</mi></mrow></msub></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$SumVar</mi><mrow><msup><mi mathvariant='normal'>$SampleSize</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$CLTVar</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$LeftLimit</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$SampleSize</mi></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>&ImaginaryI;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$SampleSize</mi></mrow></munderover><msub><mi>X</mi><mrow><mi>&ImaginaryI;</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$RightLimit</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$LeftLimit</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>1</mn></mrow><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mn>5</mn></mrow></mfrac></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$SampleSize</mi></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>&ImaginaryI;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$SampleSize</mi></mrow></munderover><msub><mi>X</mi><mrow><mi>&ImaginaryI;</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>1</mn></mrow><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mn>5</mn></mrow></mfrac></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$RightLimit</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>1</mn></mrow><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mn>5</mn></mrow></mfrac></mrow></msqrt></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ZLeft</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$ZRight</mi></mrow></mfenced></mrow></mstyle></math> where Z is the Standard normal.</p>
<p>= F($ZRight) - F($ZLeft)</p>
<p>Just look up the values in a table, or use the Normal Calculator provided.</p>@
qu.2.7.editing=useHTML@
qu.2.7.solution=@
qu.2.7.algorithm=$Q=3;
$SampleSize=range(80,200,10);
$LeftLimit=range(0.9,0.98,.01);
$RightLimit=2-$LeftLimit;
$SumVar=(1/5)*$SampleSize;
$CLTVar=(1/$SampleSize^2)*$SumVar;
$ZLeft=($LeftLimit-1)/sqrt(1/5);
$ZRight=($RightLimit-1)/sqrt(1/5);
$V = (1/5)*(1/$SampleSize);
$B = ($RightLimit-1)/sqrt($V);
$PreAns=maple("2*(stats[statevalf, cdf, normald])($B)-1");
$Ans=decimal(3,$PreAns);@
qu.2.7.uid=c19c9b71-fed6-4e6e-811a-a52340293240@
qu.2.7.info=  Course=230;
  Type=numeric;
@

qu.2.8.question=<div title="UW Statistics Bank/Continuous Probability Models/Central Limit Theorem/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.html" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a><img width="104" hspace="4" height="69" align="left" alt="" src="__BASE_URI__CPD/CLT/JarOfCoins.gif" title="Jar of Coins [IMG:JarOfCoins.gif]" />A charitable organization raises funds by asking people to put loose change in a jar. Over time, they have found that $pD% of the coins donated are dimes (10 cents), $pQ% are quarters (25 cents) and $pL% are loonies (1 dollar). What is the approximate probability that a collection box containing $nCoins coins has more than \\$$minVal in it? (4 decimals)</div>@
qu.2.8.answer.num=$Ans@
qu.2.8.answer.units=@
qu.2.8.showUnits=false@
qu.2.8.grading=toler_abs@
qu.2.8.err=0.001@
qu.2.8.negStyle=minus@
qu.2.8.numStyle=thousands scientific dollars arithmetic@
qu.2.8.mode=Numeric@
qu.2.8.name=02. P($>amount) for coins@
qu.2.8.comment=<p>Note that the value in dollars of the i'th coin is a random variable, say <em><font size="3" face="Times New Roman">X<sub>i</sub></font></em> which has probability function:</p>
<table cellspacing="0" cellpadding="3" bordercolor="#111111" border="1" style="border-collapse: collapse;" id="AutoNumber1">
    <tbody>
        <tr>
            <td><em><font size="3" face="Times New Roman">x</font></em></td>
            <td><font size="3" face="Times New Roman">0.10</font></td>
            <td><font size="3" face="Times New Roman">0.25</font></td>
            <td><font size="3" face="Times New Roman">1.00</font></td>
        </tr>
        <tr>
            <td><font size="3" face="Times New Roman"><em>f</em>(<em>x</em>)</font></td>
            <td><font size="3" face="Times New Roman">$pDime</font></td>
            <td><font size="3" face="Times New Roman">$pQuarter</font></td>
            <td><font size="3" face="Times New Roman">$pLoonie</font></td>
        </tr>
    </tbody>
</table>
<p><br />
To apply the <em>Central Limit Theorem</em> we need:<br />
<font size="3" face="Times New Roman">E(<em>X<sub>i</sub></em>)=0.1($pDime) + 0.25($pQuarter) + 1($pLoonie) = $EXi<br />
E(<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>X</mi><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>) = (0.1)<sup>2</sup>($pLoonie) + (0.25)<sup>2</sup>($pQuarter) + 1<sup>2</sup>($pLoonie) = $EXi2</font><br />
<font size="3" face="Times New Roman">Var(<em>X<sub>i</sub></em>) = $EXi2 - (EXi)<sup>2</sup> = $VarXi</font><br />
<font size="3" face="Times New Roman">SD(<em>X<sub>i</sub></em>) = $SDXi</font></p>
<p>If Y is the sum of the coin values, then we have: <br />
<font size="3" face="Times New Roman"><em>Y</em> ~ N($nCoins*$EXi,$nCoins*$VarXi) = N(</font> <font size="3" face="Times New Roman">$SEXi,$SVar)</font>.</p>
<p>Using the CLT:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><munderover><mi>&Sum;</mi><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mi mathvariant='normal'>$nCoins</mi></munderover><msub><mi>X</mi><mi>i</mi></msub></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$minVal</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><munderover><mi>&Sum;</mi><mrow><mi>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mi mathvariant='normal'>$nCoins</mi></munderover><msub><mi>X</mi><mi>i</mi></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$SEXi</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$SVar</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mfrac><mrow><mi mathvariant='normal'>$minVal</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$SEXi</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$SVar</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$minValStandard</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'></mo><mrow><mo lspace='0.0em' rspace='0.0em'>&#8804;</mo></mrow><mi mathvariant='normal'>$minValStandard</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$FPreans</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math><br />
(Using a table or the calculator provided.)</p>
<p>&nbsp;</p>@
qu.2.8.editing=useHTML@
qu.2.8.solution=@
qu.2.8.algorithm=$Q=2;
$nCoins=range(100,400,25);
$pDime=decimal(2,range(0.05,0.65,.05));
$pQuarter=decimal(2,range(0.1,0.85-$pDime,.05));
$pLoonie=1-$pDime-$pQuarter;
$pD=100*$pDime;
$pQ=100*$pQuarter;
$pL=100*$pLoonie;
$EXi = 0.1*$pDime+0.25*$pQuarter+$pLoonie;
$SEXi=decimal(2,$nCoins*$EXi);
$EXi2= 0.1^2*$pDime+0.25^2*$pQuarter+$pLoonie;
$VarXi = $EXi2-($EXi)^2;
$SVar=decimal(2,$nCoins*$VarXi);
$SDXi = decimal(3,sqrt($VarXi));
$minVal=range(int(0.9*$SEXi),int(1.2*$SEXi));
$minValStandard=($minVal - $EXi*$nCoins)/sqrt($SVar);
$Preans = maple("(stats[statevalf, cdf, normald])($minValStandard)");
$FPreans = decimal(3,$Preans);
$Ans = decimal(3,1-$Preans);@
qu.2.8.uid=e069a897-64f2-4869-9fcf-22121e518481@
qu.2.8.info=  Course=230;
  Type=numeric;
@

qu.2.9.question=<div title="UW Statistics Bank/Continuous Probability Models/Central Limit Theorem/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.html" target="Popup"><img border="0" align="$CalcAlign" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>People who have been in contact with a carrier of a disease, have a $PD% chance of contracting the disease. Suppose that the carrier of the diseases may have infected a school with $N people. Find the approximate probability that at least $EX people will contract the disease. (3 decimals)</div>@
qu.2.9.answer.num=$Ans@
qu.2.9.answer.units=@
qu.2.9.showUnits=false@
qu.2.9.grading=toler_abs@
qu.2.9.err=.01@
qu.2.9.negStyle=minus@
qu.2.9.numStyle=thousands scientific dollars arithmetic@
qu.2.9.mode=Numeric@
qu.2.9.name=07. P(Getting disease)@
qu.2.9.comment=<p><strong>The correct answer is $Ans .</strong></p>
<p>Let X be the random variable representing the number of students infected. Then X ~ Bin($N,$P). Use the Normal Approximation to the Binomial with p = $P and n = $N . The Standard Normal approximation then is:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>W</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$N</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$P</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$P</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$Q</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$PN</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$NPQ</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math> so</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi mathvariant='normal'>$EX</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$EX</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>W</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$EX</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$PN</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$NPQ</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>W</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$W</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.2.9.editing=useHTML@
qu.2.9.solution=@
qu.2.9.algorithm=$Q=7;
$P=decimal(2,range(0.15,0.75,.05));
$OneMP=1-$P;
$PD=100*$P;
$N=range(400,800,25);
$PN=$P*$N;
$EX=range(int(4*$PN/5),int(6*$PN/5),5);
$NPQ=$N*$P*$OneMP;
$W=($EX-$PN)/sqrt($NPQ);
$PreAns=1-maple("(stats[statevalf, cdf, normald])($W)");
$Ans=decimal(3,$PreAns);
$CalcAlign=switch(rint(2),"Left","Right");@
qu.2.9.uid=9a652b6d-e461-49cf-8d15-499ad40370dc@
qu.2.9.info=  Course=230;
  Type=numeric;
@

qu.3.topic=Exponential Distribution@

qu.3.1.mode=Multiple Choice@
qu.3.1.name=04. CDF@
qu.3.1.comment=<p>The cdf is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi>x</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>z</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>z</mi></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi>z</mi></mrow></munderover><mi mathvariant='normal'>$Lambda</mi><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>z</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msubsup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>z</mi></mrow></msup></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mi>x</mi></mrow></msubsup></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p>=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>x</mi></mrow></msup></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$Q=4;
$Lambda = range(2,7);@
qu.3.1.uid=d2b4ce79-0663-4110-9b3a-0c14eee81733@
qu.3.1.info=  Course=230;
  Type=MC;
@
qu.3.1.question=<div title="STAT230/Continuous Distributions/Exponential Distribution/Q4">Suppose X has an exponential distribution with probability density function :
<p><font size="3" face="Times New Roman"><em>f</em>(<em>x</em>)<em> = </em>$Lambda<em> e<sup>&minus;</sup></em><sup>$Lambda</sup></font><em><font size="3" face="Times New Roman"><sup> x</sup></font> </em>for <font size="3" face="Times New Roman"><em>x</em> > 0</font>. Then the cumulative distribution function of <font size="3" face="Times New Roman"><em>X</em></font> is:</p>
</div>@
qu.3.1.answer=1@
qu.3.1.choice.1=<font size="3" face="Times New Roman">1 - <em>e</em><sup>-$Lambda <em>x</em></sup> , <em>x</em> &gt; 0</font>@
qu.3.1.choice.2=<font size="3" face="Times New Roman"><em>e</em><sup>-$Lambda<em> x</em></sup> , <em>x</em> &gt; 0</font>@
qu.3.1.choice.3=<font size="3" face="Times New Roman">$Lambda <em>e</em><sup>-$Lambda <em>x</em></sup> , <em>x</em> > 0</font>@
qu.3.1.choice.4=The derivative of <font size="3" face="Times New Roman"><em>f</em>(<em>x</em>)</font>@
qu.3.1.choice.5=There is not enough given to determine this.@
qu.3.1.fixed=3,4@

qu.3.2.question=<div title="UW Statistics Bank/Continuous Distributions/Exponential Distribution/Q$Q">Suppose X has an exponential distribution with probability density function <font size="3" face="Times New Roman"><em>f(x) = </em>$Lambda<em> </em>e<sup>&minus;$Lambda</sup></font><em><font size="3" face="Times New Roman"><sup>\\x</sup></font> </em>for<font size="3" face="Times New Roman"> <em>x </em>> 0</font>.
<p>Then <font size="3" face="Times New Roman">P(e<sup>-<em>X</em></sup> < $Limit)</font> is (to 4 decimals):</p>
</div>@
qu.3.2.answer.num=$Ans@
qu.3.2.answer.units=@
qu.3.2.showUnits=false@
qu.3.2.grading=toler_abs@
qu.3.2.err=.001@
qu.3.2.negStyle=minus@
qu.3.2.numStyle=thousands scientific dollars arithmetic@
qu.3.2.mode=Numeric@
qu.3.2.name=05. P(exp(-X) < n)@
qu.3.2.comment=<p>Notice the following property of Exponential Distributions (for <font size="3" face="Times New Roman"><em>k</em> &ge; 0</font>) :<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>k</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>k</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi>k</mi></mrow></munderover><mi mathvariant='normal'>$Lambda</mi><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msubsup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>x</mi></mrow></msup></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>k</mi></mrow></msup></mrow></mstyle></math></p>
<p>So: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>X</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Limit</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Limit</mi></mrow></mfenced></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Limit</mi></mrow></mfenced><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Limit</mi></mrow></mfenced></mrow></mfenced></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>$Limit</mi><mrow><mi mathvariant='normal'>$Lambda</mi></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$Q=5;
$Lambda = range(2,7);
$Limit = decimal(3,range(0,0.95,.01));
$Ans=decimal(3,$Limit^$Lambda);@
qu.3.2.uid=e6761b7d-cc7c-476b-a9af-c5dac591ad18@
qu.3.2.info=  Course=230;
  Type=numeric;
@

qu.3.3.mode=Multiple Choice@
qu.3.3.name=01. P(X>n)@
qu.3.3.comment=P(X > $Ex) = 1 - P(X &#8804; $Ex)<br><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mrow><munderover accent='false' accentunder='false'><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>0</mn></mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Ex</mi></munderover><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Lambda</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8901;</mo><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>e</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Lambda</mi></mrow></mfenced><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>x</mi></mrow></msup><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&InvisibleTimes;</mo><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&DifferentialD;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>x</mi></mrow></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><msubsup superscriptshift='0' subscriptshift='0'><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='prefix' fence='true' separator='false' lspace='thinmathspace' rspace='verythinmathspace' stretchy='true' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&lpar;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>e</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Lambda</mi></mrow></mfenced><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>x</mi></mrow></msup><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='postfix' fence='true' separator='false' lspace='thinmathspace' rspace='verythinmathspace' stretchy='true' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&rpar;</mo></mrow><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>0</mn></mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Ex</mi></msubsup></mrow></mrow></math><br><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mfenced><mrow><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>-e</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Lambda</mi></mrow></mfenced><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Ex</mi></mrow></msup><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn></mrow></mfenced><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>e</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$LEx</mi></mrow></msup></mrow></math>@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$Q=1;
$Lambda = range(2,7,1);
$Ex = range(1,10,1);
$LEx=$Lambda*$Ex;
$Alt1=int($LEx/2);
$Alt2=mathml("1/$Lambda");
$Alt3=mathml("2/$Lambda");
$Alt4=$LEx+$Alt1;@
qu.3.3.uid=b01bada3-bebe-40c4-b0c5-aea56c4bb9d8@
qu.3.3.info=  Course=230;
  Type=MC;
@
qu.3.3.question=<div title="UW Statistics Bank/Continuous Probability Models/Exponential Distribution/Q$Q">Suppose X has an exponential distribution with probability density function <font size="3" face="Times New Roman"><em>f</em>(<em>x</em>)<em> = </em>$Lambda<em> e<sup>&minus;</sup></em><sup>$Lambda</sup></font><em><font size="3" face="Times New Roman"><sup> x</sup></font> </em>for <font size="3" face="Times New Roman"><em>x</em> > 0</font>. Then <font size="3" face="Times New Roman"><em>P</em>(<em>X</em> > $Ex)</font> is:

</div>@
qu.3.3.answer=1@
qu.3.3.choice.1=<font size="3" face="Times New Roman">e<sup>$LEx</sup></font>@
qu.3.3.choice.2=<font size="3" face="Times New Roman">e<sup>-$Alt1</sup></font>@
qu.3.3.choice.3=$Alt2@
qu.3.3.choice.4=$Alt3@
qu.3.3.choice.5=<font size="3" face="Times New Roman">e<sup>-$Alt4</sup></font>@
qu.3.3.fixed=@

qu.3.4.question=<div title="UW Statistics Bank/Continuous Distributions/Exponential Distribution/Q$Q">Suppose X has an exponential distribution with probability density function:
<p><font size="3" face="Times New Roman"><em>f</em>(<em>x</em>)<em> = </em>$Lambda<em> e<sup>&minus;</sup></em><sup>$Lambda</sup></font><em><font size="3" face="Times New Roman"><sup> x</sup></font> </em>for <font size="3" face="Times New Roman"><em>x</em> > 0</font>. Find <font size="3" face="Times New Roman">E(<em>X</em>)</font> (4 decimal accuracy)</p>
</div>@
qu.3.4.answer.num=1/$Lambda@
qu.3.4.answer.units=@
qu.3.4.showUnits=false@
qu.3.4.grading=toler_abs@
qu.3.4.err=.001@
qu.3.4.negStyle=minus@
qu.3.4.numStyle=thousands scientific dollars arithmetic@
qu.3.4.mode=Numeric@
qu.3.4.name=03. E(X)@
qu.3.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi>&infin;</mi></mrow></munderover><mi>xf</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'></mo><mi mathvariant='normal'>$Lambda</mi><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi>&infin;</mi></mrow></munderover><msup><mi>xe</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi mathcolor='#800080'>x</mi></mrow></mstyle></math> <br />
Such an integral is done <em>by parts</em>. Let :</p>
<p><em>u = x&nbsp; so du = dx;</em></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Lambda</mi><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>x</mi></mrow></msup><msup><mi>dx</mi><mrow><mi mathvariant='normal'></mi></mrow></msup></mrow></mstyle></math> <em>so <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>x</mi></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></em></p>
<p><em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>uv</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>vdu</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></em></p>
<p><em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>xe</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>x</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mrow></mstyle></math></em></p>
<p><em><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Lambda</mi><mi>x</mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$Lambda</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></em></p>
<p>Now evaluate this as a definite integral over the range (0,+&infin;):</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$Lambda</mi></mrow></mfrac></mrow><mrow></mrow></mstyle></math></p>@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=$Q=3;
$Lambda = range(2,7,1);
$M = range(1,10,1);
$N=range($M+1,12,1);
$AnsExp=$Lambda*($M-$N);
$Alt1=int($AnsExp/2);
$Alt2ML=mathml("1/$Lambda");
$Alt3ML=mathml("2/$Lambda");
$Alt4=$AnsExp+$Alt1;@
qu.3.4.uid=3fdd938d-347e-4e58-884a-d3d8701e3ec0@
qu.3.4.info=  Course=230;
  Type=numeric;
@

qu.3.5.mode=Multiple Choice@
qu.3.5.name=02. P(X>M | X>N)@
qu.3.5.comment=<p>P(X>$M|X>$N) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$M</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>and</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$N</mi></mrow></mfenced></mrow></mfenced></mrow><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$N</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$M</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$N</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math> (since $M > $N)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#8804;</mo><mi mathvariant='normal'>$M</mi></mrow></mfenced></mrow><mrow><mi></mi></mrow></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#8804;</mo><mi mathvariant='normal'>$N</mi></mrow></mfenced></mrow><mrow><mi></mi></mrow></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mi mathvariant='normal'>$M</mi></munderover><mi mathvariant='normal'>$Lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#8901;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi mathvariant='normal'>$Lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#8901;</mo><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mrow></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>0</mn><mi mathvariant='normal'>$N</mi></munderover><mi mathvariant='normal'>$Lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#8901;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi mathvariant='normal'>$Lambda</mi><mo lspace='0.0em' rspace='0.0em'>&#8901;</mo><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mrow></mrow></mfrac><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>&uminus0;e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi mathvariant='normal'>$M</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8901;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>&uminus0;e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi mathvariant='normal'>$N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8901;</mo><mi mathvariant='normal'>$Lambda</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mi mathvariant='normal'>$Lambda</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$M</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$N</mi></mrow></mfenced></mrow></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.3.5.editing=useHTML@
qu.3.5.solution=@
qu.3.5.algorithm=$Q=2;
$Lambda = range(2,4);
$N = range(1,10);
$M=range($N+1,$N+5-$Lambda);
$AnsExp=-$Lambda*($M-$N);
$Ans=decimal(4,exp($AnsExp));
$Alt1=int($AnsExp/2);
$Alt2=mathml("1/$Lambda");
$Alt3=mathml("2/$Lambda");
$Alt4=$AnsExp+$Alt1;@
qu.3.5.uid=2fcf97a3-5d58-4f22-930d-fd856090fca8@
qu.3.5.info=  Course=230;
  Type=MC;
@
qu.3.5.question=<div title="UW Statistics Bank/Continuous Distributions/Exponential Distribution/Q$Q">
Suppose X has an exponential distribution with probability density function:
<p><font size="3" face="Times New Roman"><em>f</em>(<em>x</em>)<em> = </em>$Lambda<em>\\e<sup>&minus;</sup></em><sup>$Lambda</sup></font><em><font size="3" face="Times New Roman"><sup>\\x</sup></font> </em>for <font size="3" face="Times New Roman"><em>x</em> > 0</font>. Then <font size="3" face="Times New Roman"><em>P</em>(<em>X </em>> $M | <em>X</em> > $N)</font> is:</p>
</div>@
qu.3.5.answer=1@
qu.3.5.choice.1=<font size="3" face="Times New Roman"><em>e</em><sup>-$AnsExp</sup></font>@
qu.3.5.choice.2=<font size="3" face="Times New Roman"><em>e</em><sup>-$Alt1</sup></font>@
qu.3.5.choice.3=$Alt2@
qu.3.5.choice.4=$Alt3@
qu.3.5.choice.5=<font size="3" face="Times New Roman"><em>e</em><sup>-$Alt4</sup></font>@
qu.3.5.fixed=@

qu.4.topic=Normal Distribution Models@

qu.4.1.mode=Inline@
qu.4.1.name=20. Boy Scouts badge@
qu.4.1.comment=<p>Let X be the number receiving their badge. Use the normal approximationn to the binomial so:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>np</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$N</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mi mathvariant='normal'>$PER</mi><mrow><mn>100</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$U</mi></mrow></mstyle></math> , <br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>np</mi><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$N</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi mathvariant='normal'>$PER</mi></mrow><mrow><mn>100</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi mathvariant='normal'>$PER</mi></mrow><mrow><mn>100</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$V</mi></mrow></mstyle></math>.</p>
<p>Then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi mathvariant='normal'>$X1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$X1</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$V</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$V</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>&nbsp; (applying the continuity correction)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$PAns</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo>or</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Ans%</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p><em>The answer w/o the continuity correction is $AnsNCC%.</em></p>
<p>&nbsp;</p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$Q=20;
$P = range(0.3,0.35,0.001);
$N = range(40,45);
$X1 = range(10,14);
$U = $P*$N;
$V = decimal(4,$N*$P*(1-$P));
$Z1=($X1-0.5-$U)/sqrt($V);
$ZNCC=($X1-$U)/sqrt($V);
$PreAns = maple("(stats[statevalf,cdf,normald]($Z1))");
$PreAnsNCC=maple("(stats[statevalf,cdf,normald]($ZNCC))");
$P1=decimal(4,$PreAns);
$PAns=1-$P1;
$Ans = decimal(4,1-$P1)*100;
$AnsNCC=decimal(4,1-$PreAnsNCC)*100;
$PER = 100*$P;@
qu.4.1.uid=84b077f1-6099-425a-b481-85cca4c08124@
qu.4.1.info=  Type=numeric;
  Course=202;
  Keyword=binomial;
  Keyword=continuity correction;
@
qu.4.1.weighting=1@
qu.4.1.numbering=alpha@
qu.4.1.part.1.name=sro_id_1@
qu.4.1.part.1.answer.units=@
qu.4.1.part.1.numStyle=thousands scientific  arithmetic@
qu.4.1.part.1.editing=useHTML@
qu.4.1.part.1.showUnits=false@
qu.4.1.part.1.err=0.5@
qu.4.1.part.1.question=(Unset)@
qu.4.1.part.1.mode=Numeric@
qu.4.1.part.1.grading=toler_abs@
qu.4.1.part.1.negStyle=minus@
qu.4.1.part.1.answer.num=$Ans@
qu.4.1.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">Of the members of a Boy Scout troop, $PER% have received their first aid badge. If $N boy scouts are selected at random, find the probability (as a percentage, 2 decimals) that $X1 or more will have the first aid badge?<p><1><span>&nbsp; %<br /></span></p></div>@

qu.4.2.mode=Multiple Choice@
qu.4.2.name=27. Parking Ticket@
qu.4.2.comment=<p>Let X be the number of tickets received.</p>
<p>This is a binomial situation with <font size="3" face="Times New Roman"><em>p</em> = $P</font> and <font size="3" face="Times New Roman"><em>n</em> = $N</font>,&nbsp;</p>
<p>so <font size="3" face="Times New Roman">Mean = <em>np</em> = $Mean</font>,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>Var</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>pn</mi><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi mathvariant='normal'>$VarX</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SD</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>Using a Normal approximation with Continuity Correction we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Limit</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>Thus, P(X &ge; $Limit) &asymp; 1 - P(Z < $Z) = $Ans.</p>
<p>&nbsp;</p>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$Q=27;
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");
$P=range(0.25,0.27,0.01);
$N = range(250,270,1);
$U = $P*$N;
$X = range(73,75,1);
$V = $N*$P(1-$P);
$Z = ($X - 0.5 -$U)/sqrt($V);
$Pr = maple("(stats[statevalf,cdf,normald])($Z)");
$Ans = decimal(4, 1-$Pr);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+$Alt1));
$Alt4=decimal(4,0.5*($Ans+$Alt2));
$VarX = $V;
$SD = sqrt($V);
$Limit = $X;
$Mean = $U;@
qu.4.2.uid=8eccfc04-3354-49d3-b036-56f32f0017e8@
qu.4.2.info=  Course=202;
  Type=MC;
  Keyword=binomial;
  Keyword=continuity correction;
@
qu.4.2.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" vspace="4" align="$Align" src="__BASE_URI__CPD/NDM/Parking$Which.gif" alt="Parking" title="Parking problem [IMG:parking$Which.gif]" />The probability of getting a parking ticket when not paying for a 2-hour period is $P. What is the probability of getting at least $X tickets if you park on $N occasions for a 2-hour period and don&rsquo;t pay?</div>@
qu.4.2.answer=1@
qu.4.2.choice.1=$Ans@
qu.4.2.choice.2=$Alt1@
qu.4.2.choice.3=$Alt2@
qu.4.2.choice.4=$Alt3@
qu.4.2.choice.5=$Alt4@
qu.4.2.fixed=@

qu.4.3.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a><img hspace="4" align="left" src="__BASE_URI__CPD/NDM/Award$Which.gif" alt="An award" title="An award [IMG:Award$Which.gif]" />Mrs. Smith's reading class can read a mean of $Mean words per minute with a standard deviation of $SD words per minute. The top $C % of the class is to receive a special award. What is the minimum number of words per minute a student would have to read in order to get the award? (Round off to an integer!)</div>@
qu.4.3.answer.num=$Ans@
qu.4.3.answer.units=@
qu.4.3.showUnits=false@
qu.4.3.grading=exact_value@
qu.4.3.negStyle=minus@
qu.4.3.numStyle=thousands scientific dollars arithmetic@
qu.4.3.mode=Numeric@
qu.4.3.name=03a. Score for top x% readers@
qu.4.3.comment=<p>Let X represent the wpm score for the class. We want x such that P(X > x) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$C</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mstyle></math> , or more usefully such that P(X < x) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$C</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Standardizing we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$C</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mrow></mstyle></math>. Use the inverse normal to find: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SD</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$Mean</mi></mrow></mstyle></math> which we round off to an integer to get the answer of $Ans</p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$Q="03a";
$Align=switch(rint(2),"Left","Right");
$Which=rint(5);
$C=range(2,5,1);
$Mean=range(150,200,5);
$SD=range(15,28,1);
$Z=maple("(stats[statevalf, icdf, normald])(1-$C/100)");
$Ans=int(0.5+$SD*$Z+$Mean);@
qu.4.3.uid=4ca09fdb-fe70-4c80-9f46-58d9f9835ff2@
qu.4.3.info=  Course=202;
  Type=numeric;
@

qu.4.4.mode=Multiple Choice@
qu.4.4.name=02b. Estimate % preemies@
qu.4.4.comment=<p>First convert the weeks to days, so we want the probability a woman gives birth in 7*$Y = $PreemDays days or less after conception. Standardize: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$PreemDays</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mrow></mstyle></math>=$Z . Now&nbsp; find P(Z < $Z) using a table or calculator.</p>@
qu.4.4.editing=useHTML@
qu.4.4.solution=@
qu.4.4.algorithm=$Q="02b";
$Which=rint(6);
$Align=switch(rint(2),"Left","Right");
$CAlign=if(eq($Align,Left),"Right","Left");
$U = range(270,275,1);
$S = range(16,18,0.1);
$Y = range(35,38,1);
$PreemDays = 7*$Y;
$Z = ($PreemDays-$U)/$S;
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$Ans = decimal(4,$P);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.4.uid=943dcc16-b5b0-4e31-ab29-7b5d8162672c@
qu.4.4.info=  Course=202;
  Type=MC;
@
qu.4.4.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=365,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:calculator.gif]" /></a><img hspace="4" align="left" src="__BASE_URI__CPD/NDM/Baby$Which.gif" alt="Baby" title="Baby [IMG:Baby$Which.gif]" />If one assumes that the gestational age is normally distributed with mean $U days and standard deviation $S days, what proportion of births would be considered pre-term (less than $Y weeks)?</div>@
qu.4.4.answer=1@
qu.4.4.choice.1=$Ans@
qu.4.4.choice.2=$Alt1@
qu.4.4.choice.3=$Alt2@
qu.4.4.choice.4=$Alt3@
qu.4.4.fixed=@

qu.4.5.mode=Multiple Choice@
qu.4.5.name=26. Basketball@
qu.4.5.comment=<p>Let X be the number of fouls sunk.</p>
<p>&nbsp;</p>
<p>Use the Normal approximation to the binomial with continuity correction.</p>
<p>Notice that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$PP</mi><mrow><mn>100</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P</mi></mrow></mstyle></math>.</p>
<p><font size="3" face="Times New Roman">Mean = <em>np</em> = $N($P) = $U</font>,</p>
<p><font size="3" face="Times New Roman">Var = <em>np</em>(1 - <em>p</em>) = $U($OMP) = $Var </font></p>
<p>so <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi mathvariant='normal'>$Var</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SD</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Z1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2</mi></mrow></mfenced></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.4.5.editing=useHTML@
qu.4.5.solution=@
qu.4.5.algorithm=$Q=26;
$Which=rint(6);
$Align=switch(rint(2),"Left","Right");
$N = range(95,115);
$P = range(0.7,0.75,0.001);
$PP=decimal(1,100*$P);
$OMP=1-$P;
$U = $P*$N;
$Var = $N*$P*(1-$P);
$SD=decimal(4,sqrt($Var));
$X1 = range(64,70);
$X2 = range(88,95);
$Z1 = decimal(4,($X1 - 0.5 - $U)/$SD);
$Z2 = decimal(4,($X2 + 0.5 - $U)/$SD);
$Ans1=$Z1;
$Ans2=$Z2;
$Alt11=decimal(4,range(1.1,1.9,0.01)*$Ans1);
$Alt12=decimal(4,range(1.1,1.9,0.01)*$Ans2);
condition:lt($Alt11,$Alt12);
$Alt21=decimal(4,range(0.5,0.9,0.01)*$Ans1);
$Alt22=decimal(4,range(0.5,0.9,0.01)*$Ans2);
condition:lt($Alt21,$Alt22);
$Alt31=decimal(4,0.5*($Ans1+$Alt11));
$Alt32=decimal(4,0.5*($Ans2+$Alt12));
condition:lt($Alt31,$Alt32);
$Alt41=decimal(4,0.5*($Ans1+$Alt21));
$Alt42=decimal(4,0.5*($Ans2+$Alt22));
condition:lt($Alt41,$Alt42);@
qu.4.5.uid=2693def7-69d5-497d-9231-4b7b1a3154c4@
qu.4.5.info=  Keyword=binomial;
  Keyword=continuity correction;
@
qu.4.5.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__CPD/NDM/Basketball$Which.gif" alt="Basketball" title="Basketball [IMG:Basketball.gif]" />A professional basketball player sinks&nbsp;$PP% of his foul shots, in the long run. If he gets&nbsp;$N tries during a season, then the probability that he sinks between&nbsp;$X1 and&nbsp;$X2 shots (inclusive) is approximately equal to:</div>@
qu.4.5.answer=5@
qu.4.5.choice.1=<i>P</i>($Alt11 &#8804; <i>Z</i> &#8804; $Alt12)@
qu.4.5.choice.2=<i>P</i>($Alt21 &#8804; <i>Z</i> &#8804; $Alt22)@
qu.4.5.choice.3=<i>P</i>($Alt31 &#8804; <i>Z</i> &#8804; $Alt32)@
qu.4.5.choice.4=<i>P</i>($Alt41 &#8804; <i>Z</i> &#8804; $Alt42)@
qu.4.5.choice.5=<i>P</i>($Z1 &#8804; <i>Z</i> &#8804; $Z2)@
qu.4.5.fixed=@

qu.4.6.mode=Multiple Choice@
qu.4.6.name=25b. Song Length@
qu.4.6.comment=<p>Let X be the mean length of the song.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P</mi></mrow></mstyle></math></p>@
qu.4.6.editing=useHTML@
qu.4.6.solution=@
qu.4.6.algorithm=$Q="25b";
$SongType=rint(3);
$Song=switch($SongType,"country & western","rock","easy listening");
$U = range(145,155);
$S = range(30,35);
$N = range(12,24);
$X = range(143,151,0.01);
$Z = decimal(4,($X-$U)/($S/sqrt($N)));
$PreP = maple("(stats[statevalf,cdf,normald])($Z)");
$P=decimal(4,$PreP);
$Ans = $P;
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.6.uid=468da561-745a-4074-a3bb-ce232dedd01c@
qu.4.6.info=  Type=MC;
  Course=202;
@
qu.4.6.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">If the length of $Song songs has mean&nbsp;$U seconds and standard deviation&nbsp;$S seconds, then the probability that a random selection of&nbsp;$N songs will have mean length of&nbsp;$X seconds or less is<a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img hspace="4" border="0" align="$CAlign" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a></div>@
qu.4.6.answer=1@
qu.4.6.choice.1=$Ans@
qu.4.6.choice.2=$Alt1@
qu.4.6.choice.3=$Alt2@
qu.4.6.choice.4=$Alt3@
qu.4.6.fixed=@

qu.4.7.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" height="75" width="50" vspace="2" align="right" title="Rocks [IMG:Rocks.gif]" alt="Rocks" src="__BASE_URI__CPD/NDM/Rocks.gif" />The mean weight of loads of rock is&nbsp;$U tons with a standard deviation of&nbsp;$S tons. If $N loads are chosen at random for a weight check, find the probability that the mean weight of those loads is NO less than $X tons (4 decimals). Assume that the variable is normally distributed.<a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img hspace="4" border="0" align="$CalcAlign" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a></div>@
qu.4.7.answer.num=$Ans@
qu.4.7.answer.units=@
qu.4.7.showUnits=false@
qu.4.7.grading=toler_abs@
qu.4.7.err=.001@
qu.4.7.negStyle=minus@
qu.4.7.numStyle=thousands scientific dollars arithmetic@
qu.4.7.mode=Numeric@
qu.4.7.name=17. Average weight of rocks@
qu.4.7.comment=<p>Let the average weight of a load be X.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P4</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.7.editing=useHTML@
qu.4.7.solution=@
qu.4.7.algorithm=$Q="17";
$U = range(46,47,0.01);
$S = range(10,15,0.01);
$N = range(5,9);
$X = range(43,45,0.01);
$Z = ($X-$U)/($S/sqrt($N));
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$P4=decimal(4,$P);
$Ans = 1-$P4;
$CalcAlign=switch(rint(2),"Left","Right");@
qu.4.7.uid=c94444a8-7422-4aaf-85ad-58f6c11fb936@
qu.4.7.info=  Type=numeric;
  Course=202;
@

qu.4.8.mode=Multiple Choice@
qu.4.8.name=07. Crocodile Length@
qu.4.8.comment=<p><br />
Let X be the crocodile length.</p>
<p>Then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mi>x</mi></mrow></msub></mrow></mrow><mrow><msub><mi>&sigma;</mi><mrow><mi>x</mi></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mrow></mstyle></math> is standard normal .</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$Limit</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mfrac><mrow><mi>$Limit</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$NormLimit</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.4.8.editing=useHTML@
qu.4.8.solution=@
qu.4.8.algorithm=$Q="07";
$Which=rint(5);
$Align=switch(rint(2),"Left","Right");
$Mean=range(11.25,13.5,0.05);
$SD=range(1.4,3.1,0.1);
$Limit=range(int($Mean),int($Mean)+2,1);
$NormLimit=decimal(3,($Limit-$Mean)/$SD);
$PreAns=1-maple("(stats[statevalf, cdf, normald])($NormLimit)");
$Ans=decimal(3,$PreAns);
$Alt1=decimal(3,range(1.1,1.3,.1)*$Ans);
$Alt2=decimal(3,range(0.5,0.85,.01)*$Ans);
$UseMe=switch(rint(2),$Alt1,$Alt2);
$Alt3=decimal(3,($Ans+$UseMe)/2);@
qu.4.8.uid=64b164af-7282-44d0-baa7-cfd5fb1d5bef@
qu.4.8.info=  Type=MC;
  Course=202;
@
qu.4.8.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a><img hspace="4" vspace="2" align="$Align" title="Crocodile [IMG:Croc$Which.gif]" alt="Crocodile" src="__BASE_URI__CPD/NDM/Croc$Which.gif" />The average length of crocodiles in a swamp is $Mean feet. If the lengths are normally distributed with a standard deviation of $SD feet, find the probability that a crocodile is more than $Limit feet long.</div>@
qu.4.8.answer=4@
qu.4.8.choice.1=$Alt1@
qu.4.8.choice.2=$Alt2@
qu.4.8.choice.3=$Alt3@
qu.4.8.choice.4=$Ans@
qu.4.8.fixed=@

qu.4.9.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" align="$Align" title="$ImgName [IMG:Catch$ImgName$Which.gif]" alt="$ImgName" src="__BASE_URI__CPD/NDM/Catch$ImgName$Which.gif" />A survey of $N $Catch fishermen found that they catch an average of $Mean kg of $Catch per day with a standard deviation of $S kg. If a random sample of $n fishermen is selected, what is the probability that their average catch is less than $X kg? ( 3 decimal accuracy.)<a href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img hspace="4" border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a></div>@
qu.4.9.maple=if ($RESPONSE<= 1) then
evalb(abs(($AnsDecimal)-$RESPONSE)<0.01) else
evalb(abs($Ans-$RESPONSE)<.1) end if;@
qu.4.9.allow2d=1@
qu.4.9.maple_answer=printf(MathML[ExportPresentation]($Ans))@
qu.4.9.type=formula@
qu.4.9.mode=Maple@
qu.4.9.name=23. Fishing Catch (P or %)@
qu.4.9.comment=<p><font size="1"><em>(Although the % version of the answer is shown, the Probability form is also accepted.)</em></font></p>
<p>Let X be the average catch (in kg). Then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$AnsDecimal</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>or</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Ans</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>%</mi></mrow></mstyle></math></p>@
qu.4.9.editing=useHTML@
qu.4.9.solution=@
qu.4.9.algorithm=$Q=23;
$C=rint(4);
$Catch=switch($C,"lobster","crab","oyster","mussel");
$Which=rint(3);
$ImgName=switch($C,"Lobster","Crab","Oyster","Mussel");
$Align=switch(rint(2),"Left","Right");
$N=range(230,480,10);
$n=range(26,38,2);
$Mean=range(30,36);
$S=range(2,6);
$X=range($Mean-1,$Mean+1,0.25);
$Z=($X-$Mean)/($S/sqrt($n));
$PreAns=maple("(stats[statevalf, cdf, normald])($Z)");
$Ans=decimal(2,100*$PreAns);
$AnsM="$Ans%";
$AnsDecimal=decimal(4,$Ans/100);@
qu.4.9.uid=0753b7b5-9ebf-4218-962b-f4710bdca08c@
qu.4.9.info=  Course=202;
  Type=Maple;
@

qu.4.10.mode=Inline@
qu.4.10.name=14a. Vehicle Age@
qu.4.10.comment=<p>Let X be the average age of vehicles. Standardize X:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Age</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p>So we can now restate the question as one involving the standard normal:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$LowerMean</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$UpperMean</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$LowerMean</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Age</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$UpperMean</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Age</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>
<p>= P(Z<$UpperZ) - P(Z<$LowerZ) = $UpperP - $LowerP = $AnsDecimal&nbsp; or $Ans%</p>@
qu.4.10.editing=useHTML@
qu.4.10.solution=@
qu.4.10.algorithm=$Q="14a";
$Age=range(95,100);
$SS=range(30,50);
$UpperMean=range($Age+4,$Age+8);
$LowerMean=range($Age+2,$UpperMean-2);
$SD=range(12,24);
$LowerZ=decimal(4,($LowerMean-$Age)/($SD/sqrt($SS)));
$UpperZ=decimal(4,($UpperMean-$Age)/($SD/sqrt($SS)));
$UpperP=maple("(stats[statevalf, cdf, normald])($UpperZ)");
$LowerP=maple("(stats[statevalf, cdf, normald])($LowerZ)");
$AnsDecimal=decimal(4,($UpperP-$LowerP));
$Ans=decimal(2,100*$AnsDecimal);@
qu.4.10.uid=9715c78e-f8da-4fc2-9b38-9ee6387a792f@
qu.4.10.info=  Type=numeric;
  Course=202;
@
qu.4.10.weighting=1@
qu.4.10.numbering=alpha@
qu.4.10.part.1.name=sro_id_1@
qu.4.10.part.1.answer.units=@
qu.4.10.part.1.numStyle=thousands scientific  arithmetic@
qu.4.10.part.1.editing=useHTML@
qu.4.10.part.1.showUnits=false@
qu.4.10.part.1.err=0.0010@
qu.4.10.part.1.question=(Unset)@
qu.4.10.part.1.mode=Numeric@
qu.4.10.part.1.grading=toler_abs@
qu.4.10.part.1.negStyle=minus@
qu.4.10.part.1.answer.num=$Ans@
qu.4.10.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><p>The average age of a vehicle registered in Canada is about $Age months. If a random sample of $SS vehicles is selected, find the probability that the mean of their age is between $LowerMean and $UpperMean months. Assume the standard deviation for the population is $SD.</p><p><span>&nbsp;</span><1><span>&nbsp;</span>(4 decimal accuracy)<a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img hspace="4" border="0" align="$CAlign" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a></p></div>@

qu.4.11.mode=Multiple Choice@
qu.4.11.name=16. Mortgage Payments@
qu.4.11.comment=<p>Let X represent the average mortgage payment.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$Upper</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Upper</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$Upper</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi mathvariant='normal'>$PreAns</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.11.editing=useHTML@
qu.4.11.solution=@
qu.4.11.algorithm=$Q="16";
$Mean=range(695,765,1);
$SD=range(250,421,1);
$SS=range(65,135,5);
$Upper=$Mean+range(15,60,1);
$PAns=maple("(stats[statevalf, cdf, normald])(($Upper-$Mean)/($SD/sqrt($SS)))");
$PreAns=decimal(4,$PAns);
$Ans=1-$PreAns;
$Alt1=decimal(4,range(0.2,0.8,0.01)*$Ans);
$Alt2=decimal(4,$Ans+range(0.2,0.8,0.01)*(1-$Ans));
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));
$CalcAlign=switch(rint(2),"Left","Right");@
qu.4.11.uid=5101901d-7a1e-4ec4-a66a-333e35cf9e99@
qu.4.11.info=  Type=MC;
  Course=202;
@
qu.4.11.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img hspace="4" border="0" align="$CalcAlign" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>The average monthly mortgage payment for recent home buyers in Winnipeg is &mu; = \\$$Mean, with standard deviation of &sigma; = \\$$SD. A random sample of $SS recent home buyers is selected. The approximate probability that their average monthly mortgage payment will be more than \\$$Upper is:</div>@
qu.4.11.answer=1@
qu.4.11.choice.1=$Ans@
qu.4.11.choice.2=$Alt1@
qu.4.11.choice.3=$Alt2@
qu.4.11.choice.4=$Alt3@
qu.4.11.fixed=@

qu.4.12.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" align="$Align" title="Baseball [IMG:Baseball$Which.gif]" alt="Baseball" src="__BASE_URI__CPD/NDM/Baseball$Which.gif" />If a baseball player's batting average is $P or $PER%, find the probability that the player will have a bad season and only score at most $X2 hits in $N times at bat? (4 decimals, answer as a probability, not a %)</div>@
qu.4.12.answer.num=$Ans@
qu.4.12.answer.units=@
qu.4.12.showUnits=false@
qu.4.12.grading=toler_abs@
qu.4.12.err=.001@
qu.4.12.negStyle=minus@
qu.4.12.numStyle=thousands scientific dollars arithmetic@
qu.4.12.mode=Numeric@
qu.4.12.name=12. Batter@
qu.4.12.comment=<p>Let X be the number of hits in $N at-bats. We want P(X &le; $X2)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.12.editing=useHTML@
qu.4.12.solution=@
qu.4.12.algorithm=$Q="12";
$P = range(0.3,0.35,0.001);
$X2 = range(60,72);
$N = range(int(3.5*$X2),4*$X2);
$U = $P*$N;
$V = $N*$P*(1-$P);
$Z2=($X2+0.5-$U)/sqrt($V);
$PreP1 = maple("(stats[statevalf,cdf,normald]($Z2))");
$Ans = decimal(4,$PreP1);
$PER = 100*$P;
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");@
qu.4.12.uid=cf011123-5377-4a5e-be23-fa4568cc6cab@
qu.4.12.info=  Type=numeric;
  Course=202;
@

qu.4.13.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>At a large department store, the average number of years of employment for a cashier is $U with a standard deviation of $S years. If an employee is picked at random, what is the probability that the employee has worked at the store for over $X years? Express your answer as a % with 3 decimal accuracy (see the Hint for more details.)</div>@
qu.4.13.answer.num=$Ans@
qu.4.13.answer.units=@
qu.4.13.showUnits=false@
qu.4.13.grading=toler_abs@
qu.4.13.err=0.01@
qu.4.13.negStyle=minus@
qu.4.13.numStyle=thousands scientific dollars arithmetic@
qu.4.13.mode=Numeric@
qu.4.13.name=05a. P(worked>X years)@
qu.4.13.comment=<p>Let X be the number of years worked by an employee.</p>
<p>We want P(X > $X) = 1 - P(X < $X)<br />
= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
= 1 - $P&nbsp; <br />
To get the answer in % multiply this by 100.</p>@
qu.4.13.editing=useHTML@
qu.4.13.hint.1=Suppose the answer was a Probability of 0.123 . Then answer as 12.3 . The following would be <strong>wrong</strong>:&nbsp; 0.123, 12.3%, 0.123% . Another example in the second hint.@
qu.4.13.hint.2=You calculate the probability as 0.87658 . Type in 87.66 (=100*0.87658 rounded to 2 decimals) as your answer. The following would be <strong>wrong</strong>: 87.66%, 87.65, 0.88, 0.8865, 0.8866@
qu.4.13.solution=@
qu.4.13.algorithm=$Q="05a";
$X = range(10,15,1);
$U = range(7,9,0.1);
$S = range(2.0,2.5,0.1);
$Z = ($X-$U)/$S;
$PreP = maple("(stats[statevalf,cdf,normald])($Z)");
$P=decimal(4,$PreP);
$N = 1-$P;
$Ans = decimal(2,100*$N);@
qu.4.13.uid=c9972e37-f92e-4c2b-923a-b2e5e5c2616d@
qu.4.13.info=  Course=202;
  Type=numeric;
@

qu.4.14.mode=Multiple Choice@
qu.4.14.name=14b. Vehicle Age@
qu.4.14.comment=<p>Let X be the average age of vehicles. Standardize X:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Age</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>
<p>So we can now restate the question as one involving the standard normal:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$LowerMean</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$UpperMean</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$LowerMean</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Age</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$UpperMean</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Age</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>
<p>= P(Z<$UpperZ) - P(Z<$LowerZ) = $UpperP - $LowerP = $PAns&nbsp; or $Ans%</p>@
qu.4.14.editing=useHTML@
qu.4.14.solution=@
qu.4.14.algorithm=$Q="14b";
$Age=range(95,100);
$SS=range(30,50);
$UpperMean=range($Age+4,$Age+8);
$LowerMean=range($Age+2,$UpperMean-2);
$SD=range(12,24);
$LowerZ=decimal(4,($LowerMean-$Age)/($SD/sqrt($SS)));
$UpperZ=decimal(4,($UpperMean-$Age)/($SD/sqrt($SS)));
$UpperP=maple("(stats[statevalf, cdf, normald])($UpperZ)");
$LowerP=maple("(stats[statevalf, cdf, normald])($LowerZ)");
$PAns=decimal(4,($UpperP-$LowerP));
$Ans=decimal(2,100*$PAns);
$Alt1=decimal(2,range(0.4,0.8,0.01)*$Ans);
$Alt2=decimal(2,range(1.2,1.8,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.14.uid=04c33a6d-3468-44fd-bc62-386e8c320f3a@
qu.4.14.info=  Type=MC;
  Course=202;
@
qu.4.14.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">The average age of a vehicle registered in Canada is about $Age months. If a random sample of $SS vehicles is selected, find the probability that the mean of their age is between $LowerMean and $UpperMean months. Assume the standard deviation for the population is $SD.<a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img hspace="4" border="0" align="$CAlign" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a></div>@
qu.4.14.answer=1@
qu.4.14.choice.1=$Ans%@
qu.4.14.choice.2=$Alt1%@
qu.4.14.choice.3=$Alt2%@
qu.4.14.choice.4=$Alt3%@
qu.4.14.fixed=@

qu.4.15.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a href="__BASE_URI__Tools/NormalCalculator.html" target="Popup" onclick="window.open(this.href,this.target,'height=130,width=340')"><img border="0" align="$AlignCalc" src="__BASE_URI__Tools/Calculator.gif" alt="Click to open the Quick Normal Calculator" title="Click to open the Quick Normal Calculator [IMG:Calculator.gif]" /></a><img hspace="4" align="$AlignPic" alt="$ImgName" title="$ImgName  [IMG:$ImgName$WhichImg.gif]" src="__BASE_URI__CPD/NDM/$ImgName$WhichImg.gif" />The mean $What of $Of\\s $Where is $M $WhatUnit, with a standard deviation of $SD. Assuming a normal distribution, the probability that a randomly chosen $Of will have $What less than $Ex $WhatUnit is: (Answer to 4 decimal accuracy.)</div>@
qu.4.15.answer.num=$Ans@
qu.4.15.answer.units=@
qu.4.15.showUnits=false@
qu.4.15.grading=toler_abs@
qu.4.15.err=.001@
qu.4.15.negStyle=minus@
qu.4.15.numStyle=thousands scientific dollars arithmetic@
qu.4.15.mode=Numeric@
qu.4.15.name=01. All-in-One@
qu.4.15.comment=<p><strong>Correct answer: $Ans</strong></p>
<p>Let X be the random variable measuring $What of $Of\\s $Where. Then &mu;<sub>X</sub> = $M $WhatUnit, and &sigma;<sub>X</sub> = $SD. To transform to the standard normal set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><msub><mi>&#956;</mi><mi>X</mi></msub></mrow></mrow><mrow><msub><mi>&#963;</mi><mi>X</mi></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mi mathvariant='normal'>$M</mi></mrow><mi mathvariant='normal'>$SD</mi></mfrac></mrow></mstyle></math>. Now the probability that a single (randomly chosen) $Of has $What less than $Ex $WhatUnit&nbsp; is: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>P</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>X</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&le;</mo></mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Ex</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>P</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>Z</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&le;</mo></mrow><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Ex</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&#8722;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$M</mi></mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$SD</mi></mfrac></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>P</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>Z</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&le;</mo></mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$SEx</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$Ans</mi></mrow></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math></p>@
qu.4.15.editing=useHTML@
qu.4.15.solution=@
qu.4.15.algorithm=$Q="01";
$nc=6;
$Choice=rint($nc);
$ImgName=switch($Choice,"House","Doctor","Nails","Cat","Typist","Train");
$AL=rint(2);
$AlignPic=switch($AL,"Left","Right");
$AlignCalc=switch(1-$AL,"Left","Right");
$WhichImg=rint(4);
$What=switch($Choice,"income","age","length","weight","keyboard speed","speed");
$WhatUnit=switch($Choice,"dollars","years","mm","kg","characters per minute","km/hr");
$Of=switch($Choice,"household","doctor","nail","cat","office worker","train");
$Where=switch($Choice,"in a certain city","in a certain hospital","in a production run","in a vet clinic","in an office","on a railroad");
$LowMean=switch($Choice,20000,39,24.5,2,30,25);
$UpperMean=switch($Choice,35000,54,26.2,10,120,100);
$MeanInc=switch($Choice,1000,1,0.01,0.1,1,2);
$M=range($LowMean,$UpperMean,$MeanInc);
$LowSD=switch($Choice,2000,2.5,0.2,0.5,4,5);
$UpperSD=switch($Choice,4000,5.5,0.8,1,8,10);
$SDInc=switch($Choice,100,0.5,0.01);
$SD=range($LowSD,$UpperSD,$SDInc);
$EXLow=switch($Choice,0.75*$M,$M-int($SD),0.90*$M,0.85*$M,0.85*$M,0.85*$M);
$Ex=range($EXLow,$M-$MeanInc,$MeanInc);
$SEx = ($Ex - $M)/$SD;
$PreAns = maple("(stats[statevalf, cdf, normald])($SEx)");
$Ans=decimal(4,$PreAns);@
qu.4.15.uid=a899dae5-9766-4a46-b13b-916acd3d14cf@
qu.4.15.info=  Type=numeric;
@

qu.4.16.mode=Multiple Choice@
qu.4.16.name=11. Assembly time@
qu.4.16.comment=<p>With mean = $U and SD = $S we can standardize this problem:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.16.editing=useHTML@
qu.4.16.solution=@
qu.4.16.algorithm=$Q="11";
$Assemble=switch(rint(4),"an electronic component","a boxing glove","a thanksgiving centrepiece","a tablet computer case");
$U = range(10,11,0.1);
$S = range(3.5,4,0.01);
$X = range(5,20);
$N = range(9,11);
$U1 = $U*$N;
$S1 = $S*sqrt($N);
$Z = decimal(4,($X-$U)/$S);
$PreP = maple ("(stats[statevalf,cdf,normald])($Z)");
$P=decimal(4,$PreP);
$Ans = decimal(4,1-$P);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.16.uid=2c66501a-a8d3-4c3e-9081-7e1217b16720@
qu.4.16.info=  Type=MC;
  Course=202;
@
qu.4.16.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">The time required to assemble $Assemble is normally distributed with a mean of&nbsp;$U minutes and a standard deviation of&nbsp;$S min. Find the probability that assembly of the components takes more than $X minutes.</div>@
qu.4.16.answer=1@
qu.4.16.choice.1=$Ans@
qu.4.16.choice.2=$Alt1@
qu.4.16.choice.3=$Alt2@
qu.4.16.choice.4=$Alt3@
qu.4.16.fixed=@

qu.4.17.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">If the length of $Song songs has mean&nbsp;$U seconds and standard deviation&nbsp;$S seconds, then the probability that a random selection of&nbsp;$N songs will have mean length of&nbsp;$X seconds or less is (4 decimal accuracy): <a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm"><img hspace="4" border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a></div>@
qu.4.17.answer.num=$Ans@
qu.4.17.answer.units=@
qu.4.17.showUnits=false@
qu.4.17.grading=toler_abs@
qu.4.17.err=.001@
qu.4.17.negStyle=minus@
qu.4.17.numStyle=thousands scientific dollars arithmetic@
qu.4.17.mode=Numeric@
qu.4.17.name=25a. Song Length@
qu.4.17.comment=<p>Let X be the mean length of the song.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P</mi></mrow></mstyle></math></p>@
qu.4.17.editing=useHTML@
qu.4.17.solution=@
qu.4.17.algorithm=$Q="25a";
$SongType=rint(3);
$Song=switch($SongType,"country & western","rock","easy listening");
$U = range(145,155);
$S = range(30,35);
$N = range(12,24);
$X = range(143,151,0.01);
$Z = decimal(4,($X-$U)/($S/sqrt($N)));
$PreP = maple("(stats[statevalf,cdf,normald])($Z)");
$P=decimal(4,$PreP);
$Ans = $P;@
qu.4.17.uid=79f78fca-c6c9-4bce-a204-d8ac271cd2ad@
qu.4.17.info=  Type=numeric;
  Course=202;
@

qu.4.18.mode=Multiple Choice@
qu.4.18.name=22. Sand Dollar diameters@
qu.4.18.comment=<p>Let X be the average diameter of the sand dollars. Then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$PD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.18.editing=useHTML@
qu.4.18.solution=@
qu.4.18.algorithm=$Q="22";
$U = range(4,4.3,0.01);
$S = range(1.5,1.8,0.01);
$N = range(11,15);
$X = range(3.8,4.1,0.001);
$Z = ($X-$U)/($S/sqrt($N));
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$PD=decimal(4,$P);
$Ans = decimal(4,1-$P);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.18.uid=0a63873f-f636-42d6-aa3e-81397cedaf4a@
qu.4.18.info=  Type=MC;
  Course=202;
@
qu.4.18.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img width="50" vspace="4" hspace="4" height="49" align="left" src="__BASE_URI__CPD/NDM/SandDollar.gif" alt="Sand Dollar" title="Sand dollar [IMG:SandDollar.gif]" />The average diameter of sand dollars on a certain island is $U centimeters with a standard deviation of $S centimeters. If $N sand dollars are chosen at random for a collection, find the probability that the average diameter of those sand dollars is more than $X centimeters. Assume that the variable is normally distributed.<a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img hspace="4" border="0" align="$CAlign" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a></div>@
qu.4.18.answer=1@
qu.4.18.choice.1=$Ans@
qu.4.18.choice.2=$Alt1@
qu.4.18.choice.3=$Alt2@
qu.4.18.choice.4=$Alt3@
qu.4.18.fixed=@

qu.4.19.mode=Inline@
qu.4.19.name=09a. P(Earns > x$)@
qu.4.19.comment=<p align="center"><font size="1"><em>(Although the probability form of the answer is shown, the % form is also accepted)</em></font></p><p>&nbsp;</p>
<p>Let X be the wage and Z the standard normal.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$XF</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$UF</mi></mrow><mrow><mi mathvariant='normal'>$SF</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mfrac><mrow><mi mathvariant='normal'>$XF</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$UF</mi></mrow><mrow><mi mathvariant='normal'>$SF</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$AnsDecimal</mi></mrow></mstyle></math> or $Ans %</p>@
qu.4.19.editing=useHTML@
qu.4.19.solution=@
qu.4.19.algorithm=$Q="09a";
$U = range(6.0,6.5,0.01);
$UF=numfmt("#.00",$U);
$S = range(0.4,0.5,0.05);
$SF=numfmt("#.00",$S);
$X = range(6.0,7.0,0.05);
$XF=numfmt("#.00",$X);
$Z = decimal(4,($X - $U)/$S);
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$AnsDecimal=decimal(4,1-$P);
$Ans = decimal(4,1-$P)*100;@
qu.4.19.uid=e6e9af39-6fc1-447a-bf1c-74f510eed15f@
qu.4.19.info=  Type=maple;
  Course=202;
@
qu.4.19.weighting=1@
qu.4.19.numbering=alpha@
qu.4.19.part.1.name=sro_id_1@
qu.4.19.part.1.maple_answer=printf(MathML[ExportPresentation]($AnsDecimal))@
qu.4.19.part.1.editing=useHTML@
qu.4.19.part.1.question=(Unset)@
qu.4.19.part.1.libname=@
qu.4.19.part.1.mode=Maple@
qu.4.19.part.1.allow2d=1@
qu.4.19.part.1.plot=@
qu.4.19.part.1.maple=if ($RESPONSE <=1) then
evalb(abs($AnsDecimal-$RESPONSE)<0.001) else evalb(abs($Ans-$RESPONSE)<0.1) end if;@
qu.4.19.part.1.type=formula@
qu.4.19.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>The average hourly wage of workers at a fast food restaurant is \\$$UF/hr with a standard deviation of&nbsp; \\$$SF/hr. Assume that the distribution is normally distributed. If a worker at this fast food restaurant is selected at random, what is the probability that the worker earns more than \\$$XF/hr? (4 decimal accuracy)<br /><br />&nbsp; <1><span>&nbsp;<br /></span><p>&nbsp;</p></div>@

qu.4.20.mode=Multiple Choice@
qu.4.20.name=06. Gas consumption@
qu.4.20.comment=<p>First standardize the two values given:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>Z</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>Z</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$X2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z2</mi></mrow></mstyle></math></p>
<p>We want <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><msub><mi>Z</mi><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><msub><mi>Z</mi><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.20.editing=useHTML@
qu.4.20.solution=@
qu.4.20.algorithm=$Q="06";
$S = range(0.1,0.3,0.01);
$U = range(7.8,8.5,0.01);
$X1 = range(7.9,8.1,0.01);
$X2 = range(8.3,8.5,0.01);
$Z1 = ($X1-$U)/$S;
$Z2 = ($X2-$U)/$S;
$M = maple("A:=(stats[statevalf,cdf,normald])($Z1);
B:=(stats[statevalf,cdf,normald])($Z2);
A,B;
");
$P1=switch(0,$M);
$P2=switch(1,$M);
$Ans=decimal(4,$P2-$P1);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.20.uid=658ee6c1-191f-433c-9c1d-5afcd2a6a178@
qu.4.20.info=  Type=MC;
  Course=202;
@
qu.4.20.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>The average gas consumption of a certain model car is&nbsp;$U litres/100km. If the gas consumption is normally distributed with a standard deviation of $S litres/100km, find the probability that a car has a gas consumption between&nbsp;$X1 and $X2 litres/100km..</div>@
qu.4.20.answer=1@
qu.4.20.choice.1=$Ans@
qu.4.20.choice.2=$Alt1@
qu.4.20.choice.3=$Alt2@
qu.4.20.choice.4=$Alt3@
qu.4.20.fixed=@

qu.4.21.mode=Multiple Choice@
qu.4.21.name=29. Deer ticks@
qu.4.21.comment=<p>Let X be the number of deer infected.</p>
<p>X is binomial with p = $p, n = $n and</p>
<p>Mean = np = $Mean, Var = np(1 - p) = $Var, so SD = $SD .</p>
<p>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.4.21.editing=useHTML@
qu.4.21.solution=@
qu.4.21.algorithm=$Q=29;
$p = range(0.55,0.7,0.001);
$PER = 100*$p;
$n = range(290,305);
$X = range(182,193);
$Mean = $p*$n;
$Var = decimal(4,$n*$p*(1-$p));
$SD=decimal(4,sqrt($Var));
$Z=decimal(4,($X+0.5-$Mean)/$SD);
$PrePZ = maple("(stats[statevalf,cdf,normald]($Z))");
$PZ=decimal(4,$PrePZ);
$Ans = decimal(4,$PZ);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+$Alt1));
$Alt4=decimal(4,0.5*($Ans+$Alt2));@
qu.4.21.uid=913cb1b3-2090-4e05-96ca-31a1659857cd@
qu.4.21.info=  Keyword=binomial;
  Keyword=continuity correction;
@
qu.4.21.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">A biologist estimates that $PER% of deer in the region carry a certain type of tick. For a  sample of $n deer selected at random, what is the chance that $X or fewer deer  have this tick?</div>@
qu.4.21.answer=1@
qu.4.21.choice.1=$Ans@
qu.4.21.choice.2=$Alt1@
qu.4.21.choice.3=$Alt2@
qu.4.21.choice.4=$Alt3@
qu.4.21.choice.5=$Alt4@
qu.4.21.fixed=@

qu.4.22.mode=Multiple Choice@
qu.4.22.name=24. Broomball@
qu.4.22.comment=<p>Use the normal approximation to the binomial with <font size="3" face="Times New Roman"><em>n</em> = $n</font> and <font size="3" face="Times New Roman"><em>p</em> = $p</font> so</p>
<p>&mu; = np = $Mean&nbsp; and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SD</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><mi>np</mi><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$SD</mi></mrow></mstyle></math> .</p>
<p>Let the number of games lost be X. Then:</p>
<p><font size="3" face="Times New Roman"><em>P</em>(<em>X</em> > $Limit) = 1 - <em>P</em>(<em>X</em> &le; $Limit)</font></p>
<p>Now standardize and apply the Continuity Correction:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$Limit</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p><em>The answer without applying the Continuity Correction is </em><font size="2" face="Times New Roman">$AnsNCC</font><em> .</em></p>
<p>&nbsp;</p>@
qu.4.22.editing=useHTML@
qu.4.22.solution=@
qu.4.22.algorithm=$Q=24;
$p=0.5;
$n=100;
$Mean=$n*$p;
$SD=sqrt($n*$p*(1-$p));
$Limit=range(int(0.55*$n),0.62*$n,1);
$Z=($Limit-$Mean+0.5)/$SD;
$PreAns=maple("(stats[statevalf, cdf, normald])($Z)");
$Ans=decimal(4,1-$PreAns);
$Alt1=decimal(4,range(0.4,0.85,0.01)*$Ans);
$Alt2=decimal(4,$Ans+range(0.4,0.85,0.01)*(1-$Ans));
$Alt3=decimal(4,0.5*($Ans+$Alt1));
$Alt4=decimal(4,0.5*($Ans+$Alt2));
$WinMaxNCC=$n-$Limit;
$ZNCC=($Limit-$Mean)/$SD;
$PreAnsNCC=maple("(stats[statevalf, cdf, normald])($ZNCC)");
$AnsNCC=decimal(4,1-$PreAnsNCC);@
qu.4.22.uid=adbbe43a-0b2c-48dd-b465-b37b2dfec488@
qu.4.22.info=  Course=202;
  Type=MC;
  Keyword=binomial;
  Keyword=continuity correction;
@
qu.4.22.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" height="81" width="104" align="right" alt="Broomball" src="__BASE_URI__CPD/NDM/BBall.gif" title="Broomball [IMG:BBall.gif]" />The National Broomball League claims to have a balanced league; that is, for any given game each team has an equal chance of winning or losing with no ties. Assuming the claim is true, what is the approximate probability that a given team will lose more than $Limit games out of the $n played?</div>@
qu.4.22.answer=1@
qu.4.22.choice.1=$Ans@
qu.4.22.choice.2=$Alt1@
qu.4.22.choice.3=$Alt2@
qu.4.22.choice.4=$Alt3@
qu.4.22.choice.5=$Alt4@
qu.4.22.fixed=@

qu.4.23.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a><img hspace="4" align="left" title="Baby [IMG:Baby$Which.gif]" alt="Baby" src="__BASE_URI__CPD/NDM/Baby$Which.gif" />If one assumes that the gestational age is normally distributed with mean $U days and standard deviation $S days, what proportion of births would be considered pre-term (less than $Y weeks)? 3 decimals please.</div>@
qu.4.23.answer.num=$Ans@
qu.4.23.answer.units=@
qu.4.23.showUnits=false@
qu.4.23.grading=toler_abs@
qu.4.23.err=0.01@
qu.4.23.negStyle=minus@
qu.4.23.numStyle=thousands scientific dollars arithmetic@
qu.4.23.mode=Numeric@
qu.4.23.name=02a. Estimate % preemies@
qu.4.23.comment=<p>First convert the weeks to days, so we want the probability a woman gives birth in 7*$Y = $PreemDays days or less after conception. Standardize: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$PreemDays</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mrow></mstyle></math>= $Z&nbsp; and find P(Z < $Z) using a Normal table or calculator.</p>@
qu.4.23.editing=useHTML@
qu.4.23.solution=@
qu.4.23.algorithm=$Q="02a";
$Which=rint(6);
$Align=switch(rint(2),"Left","Right");
$CAlign=if(eq($Align,Left),"Right","Left");
$U = range(270,275);
$S = range(16,18,0.1);
$Y = range(35,38);
$PreemDays = 7*$Y;
$Z = ($PreemDays-$U)/$S;
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$Ans = decimal(4,$P);@
qu.4.23.uid=2dd2ce86-2b35-48ba-8e05-729860c37fd6@
qu.4.23.info=  Course=202;
  Type=numeric;
@

qu.4.24.mode=Multiple Choice@
qu.4.24.name=30. Radio Callers@
qu.4.24.comment=<p>Let X be the number of callers who are $Who.</p>
<p>This is a binomial distribution with p = $p, n = $n,</p>
<p>Mean = np = $Mean and Var = np(1 - p) = $Var, so SD = $SD.</p>
<p>&nbsp;</p>
<p>Using a Normal Distribution with continuity correction to approximate, we have:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi mathvariant='normal'>$X1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.4.24.editing=useHTML@
qu.4.24.solution=@
qu.4.24.algorithm=$Q=30;
$Who=switch(rint(4),"men","women","adolescents","senior citizens");
$p = range(0.3,0.35,0.001);
$n = range(190,215);
$X1 = range(47,54);
$Mean = $p*$n;
$Var = $n*$p*(1-$p);
$SD=decimal(4,sqrt($Var));
$Z1=decimal(4,($X1-0.5-$Mean)/$SD);
$PreP1 = maple("(stats[statevalf,cdf,normald]($Z1))");
$P1=decimal(4,$PreP1);
$Ans = decimal(4,1-$P1);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+$Alt1));
$Alt4=decimal(4,0.5*($Ans+$Alt2));
$PER = 100*$p;@
qu.4.24.uid=9418a816-600f-4a82-b62e-efbc48accdca@
qu.4.24.info=  Type=MC;
  Course=202;
  Keyword=binomial;
  Keyword=continuity correction;
@
qu.4.24.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">Companies are interested in the demographics of those who listen to the radio programs they sponsor. A radio station has determined that only $PER% of listeners phoning in to a morning talk program are $Who. During a particular week, $n calls are received by this program. What is the approximate probability that at least $X1 of the callers are $Who?</div>@
qu.4.24.answer=1@
qu.4.24.choice.1=$Ans@
qu.4.24.choice.2=$Alt1@
qu.4.24.choice.3=$Alt2@
qu.4.24.choice.4=$Alt3@
qu.4.24.choice.5=$Alt4@
qu.4.24.fixed=4@

qu.4.25.mode=Multiple Choice@
qu.4.25.name=03b. Score for top x% readers@
qu.4.25.comment=<p>Let X represent the wpm score for the class. We want x such that P(X > x) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$C</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mstyle></math> , or more usefully such that P(X < x) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$C</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Standardizing we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$C</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mrow></mstyle></math>. Use the inverse normal to find: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Z</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SD</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$Mean</mi></mrow></mstyle></math> which we round off to an integer to get the answer of $Ans</p>@
qu.4.25.editing=useHTML@
qu.4.25.solution=@
qu.4.25.algorithm=$Q="03b";
$Align=switch(rint(2),"Left","Right");
$Which=rint(5);
$C=range(2,5,1);
$Mean=range(150,200,5);
$SD=range(15,28,1);
$Z=maple("(stats[statevalf, icdf, normald])(1-$C/100)");
$Ans=int(0.5+$SD*$Z+$Mean);
$Alt1=int(range(1.1,1.9,0.01)*$Ans);
$Alt2=int(range(0.5,0.9,0.01)*$Ans);
$Alt3=int(0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.25.uid=de1ca664-6105-45ad-a50c-e551b5e8fcf3@
qu.4.25.info=  Course=202;
  Type=MC;
@
qu.4.25.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a><img hspace="4" align="left" src="__BASE_URI__CPD/NDM/Award$Which.gif" alt="An award" title="An award [IMG:Award$Which.gif]" />Mrs. Smith's reading class can read a mean of $Mean words per minute with a standard deviation of $SD words per minute. The top $C % of the class is to receive a special award. What is the minimum number of words per minute a student would have to read in order to get the award? (Round off to an integer!)</div>@
qu.4.25.answer=1@
qu.4.25.choice.1=$Ans@
qu.4.25.choice.2=$Alt1@
qu.4.25.choice.3=$Alt2@
qu.4.25.choice.4=$Alt3@
qu.4.25.fixed=@

qu.4.26.mode=Multiple Choice@
qu.4.26.name=31. Politician canvassing@
qu.4.26.comment=<p>This is a binomial distribution which we will approximate with the normal.</p>
<p>&nbsp;</p>
<p>Notice however that we are interested in, X, the number of houseolds the politician canvasses that give the politician their support.</p>
<p><font size="3" face="Times New Roman"><em>p</em></font> = P(Invited in)*P(Given support once asked in) = <font size="3" face="Times New Roman">$P1($P2) = $p</font></p>
<p><font size="3" face="Times New Roman"><em>Mean</em> = <em>np</em> = $n($p) = $Mean and <em>Var</em> = <em>np</em>(1 - <em>p</em>) = $Var so <em>SD</em> = $SD<br />
</font></p>
<p>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi mathvariant='normal'>$X1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>&nbsp;&nbsp;&nbsp;&nbsp; <em>(applying the Continuity Correction</em>)<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.4.26.editing=useHTML@
qu.4.26.solution=@
qu.4.26.algorithm=$Q=31;
$P1 = range(0.5,0.55,0.001);
$P2 = range(0.8,0.85,0.001);
$PER = 100*$P1;
$PER2=100*$P2;
$n = range(100,105,1);
$X1 = range(45,48,1);
$p = $P1*$P2;
$Mean = $p*$n;
$Var = decimal(4,$n*$p*(1-$p));
$SD=decimal(4,sqrt($Var));
$Z1=decimal(4,($X1-0.5-$Mean)/$SD);
$PR = maple("(stats[statevalf,cdf,normald]($Z1))");
$Ans = decimal(4,1-$PR);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+$Alt1));
$Alt4=decimal(4,0.5*($Ans+$Alt2));@
qu.4.26.uid=cb417445-d488-4ea0-824b-6547231e71a8@
qu.4.26.info=  Type=MC;
  Course=202;
  Keyword=binomial;
  Keyword=continuity correction;
@
qu.4.26.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">A politician has targeted $n homes to visit during a week. From past experience, $PER percent of the households answer the bell and invite him in. Of this, $PER2 percent will agree with his policies. The approximate probability that the politician will get support from at least $X1 households during a week is:</div>@
qu.4.26.answer=1@
qu.4.26.choice.1=$Ans@
qu.4.26.choice.2=$Alt1@
qu.4.26.choice.3=$Alt2@
qu.4.26.choice.4=$Alt3@
qu.4.26.choice.5=$Alt4@
qu.4.26.fixed=@

qu.4.27.mode=Multiple Choice@
qu.4.27.name=10. Cargo@
qu.4.27.comment=<p>Let X be the average weight of the boxes. Then what is being&nbsp; asked is P(X > $UpperMean) or 1 - P(X
<title></title>
<meta name="GENERATOR" content="Microsoft FrontPage 5.0" />
<meta name="ProgId" content="FrontPage.Editor.Document" />&le; $UpperMean)</p>
<p>Standardizing: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mrow></mstyle></math> so we want<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$UpperMean</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$ZUpper</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$PZ</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.27.editing=useHTML@
qu.4.27.solution=@
qu.4.27.algorithm=$Q="10";
$SS=range(100,110,1);
$TotalUpper=range(330,340,1);
$UpperMean=decimal(4,$TotalUpper/$SS);
$Mean=range(3.1,3.2,0.01);
$SD=range(0.8,0.9,0.01);
$ZUpper=decimal(4,($UpperMean-$Mean)/($SD/sqrt($SS)));
$PrePZ=maple("(stats[statevalf, cdf, normald])($ZUpper)");
$PZ=decimal(4,$PrePZ);
$Ans=decimal(2,1-$PZ)*100;
condition:gt($Ans,10);
$Alt1=decimal(2,$Ans+range(0.5,0.9,0.01)*(100-$Ans));
$Alt2=decimal(2,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+$Alt1));
$Alt4=decimal(2,0.5*($Ans+$Alt2));@
qu.4.27.uid=96121b7d-7099-4bde-8a6c-3541ac4a41d9@
qu.4.27.info=  Type=MC;
  Course=202;
@
qu.4.27.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img width="50" hspace="4" height="50" align="right" src="__BASE_URI__Test5/NA/Airplane.gif" alt="Airplane" />Government regulations indicate that the total weight of cargo in a certain kind of airplane cannot exceed&nbsp;$TotalUpper kg. On a particular day a plane is loaded with&nbsp;$SS boxes of goods. If the weight distribution for individual boxes is normal with mean&nbsp;$Mean kg and standard deviation&nbsp;$SD kg, the probability that the regulations will NOT be met is:</div>@
qu.4.27.answer=1@
qu.4.27.choice.1=$Ans%@
qu.4.27.choice.2=$Alt1%@
qu.4.27.choice.3=$Alt2%@
qu.4.27.choice.4=$Alt3%@
qu.4.27.choice.5=$Alt4%@
qu.4.27.fixed=@

qu.4.28.mode=Multiple Choice@
qu.4.28.name=15b. Ave. Doctor's Age@
qu.4.28.comment=<p>Let X be the average doctor's age.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Upper</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$Upper</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.28.editing=useHTML@
qu.4.28.solution=@
qu.4.28.algorithm=$Q="15b";
$Mean=range(36,58,2);
$SD=range(4,8,1);
$SS=range(20,35,1);
$Upper=$Mean+range(.1,1,.1);
$Z=($Upper-$Mean)/($SD/sqrt($SS));
$PreAns=maple("(stats[statevalf, cdf, normald])($Z)");
$Ans=decimal(4,$PreAns);
$Alt1=decimal(4,range(0.4,0.8,0.01)*$Ans);
$Alt2=decimal(4,$Ans+range(.2,.8,0.01)*(1-$Ans));
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));
$CAlign=rint(2);
$AAlign=1-$CAlign;
$CalcAlign=switch($CAlign,"Left","Right");
$Align=switch($AAlign,"Left","Right");
$Which=rint(4);@
qu.4.28.uid=10551de9-1c0f-48d6-8a7d-7b383afdc9c9@
qu.4.28.info=  Type=MC;
  Course=202;
@
qu.4.28.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" align="$Align" title="Doctor [IMG:Doctor$Which.gif]" alt="Doctor" src="__BASE_URI__CPD/NDM/Doctor$Which.gif" />The average age of doctors in a certain hospital is $Mean years old with a standard deviation of $SD years. If $SS doctors are chosen at random for a committee, find the probability that the mean age of those doctors is less than $Upper years. Assume that the variable is normally distributed. <a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img hspace="4" border="0" align="$CalcAlign" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a></div>@
qu.4.28.answer=4@
qu.4.28.choice.1=$Alt2@
qu.4.28.choice.2=$Alt1@
qu.4.28.choice.3=$Alt3@
qu.4.28.choice.4=$Ans@
qu.4.28.fixed=@

qu.4.29.mode=Multiple Choice@
qu.4.29.name=13. Tire Life@
qu.4.29.comment=<p>Let X = tire lifetime. E(X) = $Mean and &sigma;<sub>X</sub> = $SD. We standardize X by setting <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mrow></mstyle></math>. Now consider a single tire, and let's find the probability that its lifetime is $Limit miles or more.&nbsp; That is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$Limit</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mfrac><mrow><mi mathvariant='normal'>$Limit</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$ZLimit</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$ZLimit</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$PZ1</mi></mrow></mstyle></math><br />
The probability that all four tires will last this long, assuming their lifespans are independent, is ($PZ1)<sup>4</sup> = $Ans .</p>@
qu.4.29.editing=useHTML@
qu.4.29.solution=@
qu.4.29.algorithm=$Q="13";
$Mean=range(39000,48000,1000);
$SD=range(1500,2750,50);
$Limit=$Mean-1000;
$ZLimit=decimal(4,($Limit-$Mean)/$SD);
$PZ=1-maple("(stats[statevalf, cdf, normald])($ZLimit)");
$PZ1=decimal(4,$PZ);
$Ans=decimal(3,$PZ1^4);
$Alt1=decimal(3,range(0.4,0.8,0.01)*$Ans);
$Alt2=decimal(3,$Ans+(1-$Ans)*range(0.4,0.8,0.01));
$Alt3=decimal(3,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.29.uid=7b47e3e1-d4ec-4360-81e9-1898f14f1581@
qu.4.29.info=  Type=MC;
  Course=202;
@
qu.4.29.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img width="154" vspace="4" hspace="4" height="85" align="right" title="Tires [IMG:Tires.gif]" alt="Tires" src="__BASE_URI__CPD/NDM/Tires.gif" />Assume that the tires sold by Olsen Tires are normally distributed with a mean life of $Mean miles and a standard deviation of $SD miles. If you were to buy 4 Olsen tires, what is the approximate probability that all four will last longer than $Limit miles?</div>@
qu.4.29.answer=1@
qu.4.29.choice.1=$Ans@
qu.4.29.choice.2=$Alt1@
qu.4.29.choice.3=$Alt2@
qu.4.29.choice.4=$Alt3@
qu.4.29.fixed=@

qu.4.30.mode=Multiple Choice@
qu.4.30.name=04. Admission Top n%@
qu.4.30.comment=<p>Let X represent the scores. We want x such that P(X > x) = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$PER</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mstyle></math> or more usefully <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$PER</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mrow></mstyle></math>. Standardizing we have <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$PER</mi><mrow><mn>100</mn></mrow></mfrac></mrow></mrow></mstyle></math>. The inverse normal tell us Z = $I, so <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$I</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SD</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$Mean</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.30.editing=useHTML@
qu.4.30.solution=@
qu.4.30.algorithm=$Q=5;
$Choose=rint(4);
$Institute=switch($Choose,"top University","prestigious Musical Academy","Veterinary Science program","ROTC Program");
$Test=switch($Choose,"SAT","musical audition","pre-vet screening test","Military Admission Test");
$Mean = range(1000,2000,100);
$SD = range(100,300,10);
$PER = range(3,5,1);
$P = $PER/100;
$Z = 1 - $P;
$I = maple("(stats[statevalf,icdf,normald])($Z)");
$Ans = decimal(0,$Mean + $I*$SD);
$Alt1=int(range(1.1,1.9,0.01)*$Ans);
$Alt2=int(range(0.5,0.9,0.01)*$Ans);
$Alt3=int(0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.30.uid=ea1b4cac-88f3-4975-8f36-8e07117a3eeb@
qu.4.30.info=  Course=202;
  Type=MC;
@
qu.4.30.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">
<a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>In order to be accepted into a $Institute, applicants must score within the top $PER% on the $Test. Given that this test has a mean of $Mean and a standard deviation of $SD, what is the lowest possible score a student needs to qualify for acceptance into the $Institute?</div>@
qu.4.30.answer=1@
qu.4.30.choice.1=$Ans@
qu.4.30.choice.2=$Alt1@
qu.4.30.choice.3=$Alt2@
qu.4.30.choice.4=$Alt3@
qu.4.30.fixed=@

qu.4.31.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" align="$Align" title="Doctor [IMG:Doctor$Which.gif]" alt="Doctor" src="__BASE_URI__CPD/NDM/Doctor$Which.gif" />The average age of doctors in a certain hospital is $Mean years old with a standard deviation of $SD years. If $SS doctors are chosen at random for a committee, find the probability that the mean age of those doctors is less than $Upper years. (4 decimal accuracy) Assume that the variable is normally distributed. <a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img hspace="4" border="0" align="$CalcAlign" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a></div>@
qu.4.31.answer.num=$Ans@
qu.4.31.answer.units=@
qu.4.31.showUnits=false@
qu.4.31.grading=toler_abs@
qu.4.31.err=.001@
qu.4.31.negStyle=minus@
qu.4.31.numStyle=thousands scientific dollars arithmetic@
qu.4.31.mode=Numeric@
qu.4.31.name=15a. Ave. Doctor's Age@
qu.4.31.comment=<p>Let X be the average doctor's age.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Upper</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$Upper</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$SD</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$SS</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.31.editing=useHTML@
qu.4.31.solution=@
qu.4.31.algorithm=$Q="15a";
$Mean=range(36,58,2);
$SD=range(4,8,1);
$SS=range(20,35,1);
$Upper=$Mean+range(.1,1,.1);
$Z=($Upper-$Mean)/($SD/sqrt($SS));
$PreAns=maple("(stats[statevalf, cdf, normald])($Z)");
$Ans=decimal(4,$PreAns);
$CAlign=rint(2);
$AAlign=1-$CAlign;
$CalcAlign=switch($CAlign,"Left","Right");
$Align=switch($AAlign,"Left","Right");
$Which=rint(4);@
qu.4.31.uid=3ef9e116-db20-4092-8ecf-ec76bf853a6b@
qu.4.31.info=  Type=numeric;
  Course=202;
@

qu.4.32.mode=Multiple Choice@
qu.4.32.name=09b. P(Earns > x$)@
qu.4.32.comment=<p>Let X be the wage and Z the standard normal.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$XF</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$UF</mi></mrow><mrow><mi mathvariant='normal'>$SF</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mfrac><mrow><mi mathvariant='normal'>$XF</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$UF</mi></mrow><mrow><mi mathvariant='normal'>$SF</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$DecAns</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> or <font size="3" face="Times New Roman">$Ans%</font></p>@
qu.4.32.editing=useHTML@
qu.4.32.solution=@
qu.4.32.algorithm=$Q="09b";
$U = range(6.0,6.5,0.01);
$UF=numfmt("#.00",$U);
$S = range(0.4,0.5,0.05);
$SF=numfmt("#.00",$S);
$X = range(6.0,7.0,0.05);
$XF=numfmt("#.00",$X);
$Z = decimal(4,($X - $U)/$S);
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$DecAns=decimal(4,1-$P);
$Ans = $DecAns*100;
$Alt1=decimal(1,$Ans+range(0.5,0.9,0.01)*(100-$Ans));
$Alt2=decimal(1,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(1,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.32.uid=585385a7-04ae-4b9b-b6dd-6b2ad079e34a@
qu.4.32.info=  Type=MC;
  Course=202;
@
qu.4.32.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>The average hourly wage of workers at a fast food restaurant is \\$$UF/hr with a standard deviation of&nbsp; \\$$SF/hr. Assume that the distribution is normally distributed. If a worker at this fast food restaurant is selected at random, what is the probability that the worker earns more than \\$$XF/hr?</div>@
qu.4.32.answer=1@
qu.4.32.choice.1=$Ans%@
qu.4.32.choice.2=$Alt1%@
qu.4.32.choice.3=$Alt2%@
qu.4.32.choice.4=$Alt3%@
qu.4.32.fixed=@

qu.4.33.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">The Canada Safety Council reported that $PER% of Canadian drivers read while driving. If $N drivers are selected at random, find the probability that exactly $X1 will admit to reading while driving? Answer as a percentage (e.g. 10 instead of 0.1) (2 decimal accuracy)</div>@
qu.4.33.answer.num=$Ans@
qu.4.33.answer.units=@
qu.4.33.showUnits=false@
qu.4.33.grading=toler_abs@
qu.4.33.err=.1@
qu.4.33.negStyle=minus@
qu.4.33.numStyle=thousands scientific dollars arithmetic@
qu.4.33.mode=Numeric@
qu.4.33.name=19. Drivers reading@
qu.4.33.comment=<p>Let the number of drivers reading be X.</p>
<p>This is a Binomial distribution, so&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>np</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$U</mi><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>np</mi><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>p</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$V</mi></mrow><mrow><mi></mi></mrow></mstyle></math>.</p>
<p>We will use the Normal Distribution approximation to the Binomial. Since the Normal is a continuous distribution, P(X = $X1) = 0 so we need to approximate using Continuity Correction techniques:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$X1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&simeq;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$V</mi></mrow></msqrt></mrow></mfrac></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$V</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$V</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$V</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z1</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P2Dec3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P1Dec3</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$PreAns</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='normal'  lspace='0.0em' rspace='0.0em'>or</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Ans%</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.4.33.editing=useHTML@
qu.4.33.solution=@
qu.4.33.algorithm=$Q=19;
$P = range(0.08,0.1,0.001);
$N = range(500,505);
$X1 = range(40,43);
$U = decimal(3,$P*$N);
$V = decimal(3,$N*$P*(1-$P));
$Z1=decimal(3,($X1-0.5-$U)/sqrt($V));
$Z2=decimal(3,($X1+0.5-$U)/sqrt($V));
$P1 = maple("(stats[statevalf,cdf,normald]($Z1))");
$P2 = maple("(stats[statevalf,cdf,normald]($Z2))");
$P1Dec3=decimal(3,$P1);
$P2Dec3=decimal(3,$P2);
$PreAns=decimal(3,$P2-$P1);
$Ans = $PreAns*100;
$PER = 100*$P;@
qu.4.33.uid=f661f4a9-9363-4b7e-ac3a-de2d706b618c@
qu.4.33.info=  Type=numeric;
  Course=202;
  Keyword=binomial;
  Keyword=continuity correction;
@

qu.4.34.mode=Multiple Choice@
qu.4.34.name=08. Tree Height@
qu.4.34.comment=<p>Let X represent tree height and Z the standard normal:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.34.editing=useHTML@
qu.4.34.solution=@
qu.4.34.algorithm=$Q="08";
$Tree=switch(rint(4),"flowering cherry","flowering linden","birch","basswood");
$U = range(3.0,3.5,0.01);
$S = range(0.5,1.2,0.01);
$X = range(3.2,3.6,0.1);
$Z = decimal(4,($X-$U)/$S);
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$Ans = decimal(4,$P);
$Alt1=decimal(4,$Ans+range(0.5,0.9,0.01)*(1-$Ans));
$Alt2=decimal(4,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(4,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.34.uid=fbe129b3-6a6c-46e8-9120-ffdfdb202796@
qu.4.34.info=  Type=MC;
  Course=202;
@
qu.4.34.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><a onclick="window.open(this.href,this.target,'height=140,width=340')" href="__BASE_URI__Tools/NormalCalculator.htm" target="Popup"><img border="0" align="right" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" alt="Quick Normal/InvNormal Calculator" src="__BASE_URI__Tools/Calculator.gif" /></a>The average height of $Tree trees in a nursery is&nbsp;$U m. If the heights are normally distributed with a standard deviation of $S m, find the probability that a tree is less than&nbsp;$X m tall.</div>@
qu.4.34.answer=1@
qu.4.34.choice.1=$Ans@
qu.4.34.choice.2=$Alt1@
qu.4.34.choice.3=$Alt2@
qu.4.34.choice.4=$Alt3@
qu.4.34.fixed=@

qu.4.35.mode=Multiple Choice@
qu.4.35.name=05b. P(worked>X years)@
qu.4.35.comment=<p>Let X be the number of years worked by an employee.</p>
<p>We want P(X > $X) = 1 - P(X < $X)<br />
= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mi mathvariant='normal'>$S</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
= 1 - $P&nbsp; <br />
To get the answer in % multiply this by 100.</p>@
qu.4.35.editing=useHTML@
qu.4.35.solution=@
qu.4.35.algorithm=$Q="05b";
$X = range(10,15,1);
$U = range(7,9,0.1);
$S = range(2.0,2.5,0.1);
$Z = ($X-$U)/$S;
$PreP = maple("(stats[statevalf,cdf,normald])($Z)");
$P=decimal(4,$PreP);
$N = 1-$P;
$Ans = decimal(2,100*$N);
$Alt1=decimal(2,$Ans+range(0.5,0.9,0.01)*(min(2*$Ans,100)-$Ans));
$Alt2=decimal(2,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.4.35.uid=508bae31-32f5-4c98-b501-bb960bbb21ba@
qu.4.35.info=  Course=202;
  Type=MC;
@
qu.4.35.question=<div title="UW Statistics Bank/Continuous Probability Models/Normal Distribution Models/Q$Q"><a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img border="0" align="right" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a>At a large department store, the average number of years of employment for a cashier is $U with a standard deviation of $S years. If an employee is picked at random, what is the probability that the employee has worked at the store for over $X years?</div>@
qu.4.35.answer=1@
qu.4.35.choice.1=$Ans%@
qu.4.35.choice.2=$Alt1%@
qu.4.35.choice.3=$Alt2%@
qu.4.35.choice.4=$Alt3%@
qu.4.35.fixed=@

qu.4.36.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" align="$Align" src="__BASE_URI__CPD/NDM/Mosquito$Which.gif" alt="Mosquito" title="Mosquito [IMG:Mosquito$Which.gif]" />The average number of mosquitoes in a stagnant pond is $U per square meter with a standard deviation of $S. If $N square meters are chosen at random for a mosquito count, find the probability that the average of those counts is more than $X mosquitoes per square meter. Assume that the variable is normally distributed. (4 decimal accuracy) <a target="Popup" href="__BASE_URI__Tools/NormalCalculator.htm" onclick="window.open(this.href,this.target,'height=140,width=340')"><img hspace="4" border="0" align="$CAlign" src="__BASE_URI__Tools/Calculator.gif" alt="Quick Normal/InvNormal Calculator" title="Quick Normal/InvNormal Calculator [IMG:Calculator.gif]" /></a></div>@
qu.4.36.answer.num=$Ans@
qu.4.36.answer.units=@
qu.4.36.showUnits=false@
qu.4.36.grading=toler_abs@
qu.4.36.err=.001@
qu.4.36.negStyle=minus@
qu.4.36.numStyle=thousands scientific dollars arithmetic@
qu.4.36.mode=Numeric@
qu.4.36.name=21. Mosquitoes@
qu.4.36.comment=<p>Let X be the average count (per m<sup>2</sup>). Then:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X</mi></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced></mrow></mstyle></math><br />
= <font size="3" face="Times New Roman">$Ans</font></p>@
qu.4.36.editing=useHTML@
qu.4.36.solution=@
qu.4.36.algorithm=$Q="21";
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");
$U = range(57,62);
$S = range(8,14);
$N = range(12,25);
$X = range(60,64);
$Z = ($X-$U)/($S/sqrt($N));
$P = maple("(stats[statevalf,cdf,normald])($Z)");
$Ans = decimal(4,1-$P);@
qu.4.36.uid=8184305e-1d26-47d6-8e38-b401c31dac10@
qu.4.36.info=  Course=202;
  Type=numeric;
@

qu.4.37.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q">Suppose at $University, $PER% of the students live in apartments. If $n students are randomly selected, then the probability that the number of them living in apartments will be between $X1 and $X2 inclusive, is (4 decimals):</div>@
qu.4.37.answer.num=$Ans@
qu.4.37.answer.units=@
qu.4.37.showUnits=false@
qu.4.37.grading=toler_abs@
qu.4.37.err=.001@
qu.4.37.negStyle=minus@
qu.4.37.numStyle=thousands scientific dollars arithmetic@
qu.4.37.mode=Numeric@
qu.4.37.name=28. Students in Residence@
qu.4.37.comment=<p>Let X be the number of students in apartments.</p>
<p>&nbsp;</p>
<p>This is a binomial distribution with <font size="3" face="Times New Roman"><em>p</em> = $p</font> and <font size="3" face="Times New Roman"><em>n</em> = $n</font>.</p>
<p><font size="3" face="Times New Roman"><em>Mean</em> = <em>np</em> = $Mean, <em>Var</em> = <em>np</em>(1 - <em>p</em>) = $Var</font>, so <font size="3" face="Times New Roman"><em>SD</em> = $SD</font></p>
<p>&nbsp;</p>
<p>Using the normal approximation we continuity correction we have:<font size="3" face="Times New Roman"><em><br />
</em></font></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$X2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$X1</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfrac><mrow><mi mathvariant='normal'>$X2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0.5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Mean</mi></mrow><mrow><mi mathvariant='normal'>$SD</mi></mrow></mfrac></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$Z2</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z1</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$P2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$P1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.4.37.editing=useHTML@
qu.4.37.solution=@
qu.4.37.algorithm=$Q=28;
$University=switch(rint(4),"the University of Manitoba","the University of Western Ontario","Conestoga College","Georgian College");
$p = range(0.3,0.35,0.001);
$n = range(175,185);
$X1 = range(49,55);
$X2 = range(62,69);
$Mean = $p*$n;
$Var = decimal(4,$n*$p*(1-$p));
$SD=decimal(4,sqrt($Var));
$Z1=decimal(4,($X1-0.5-$Mean)/$SD);
$Z2=decimal(4,($X2+0.5-$Mean)/$SD);
$Pre1 = maple("(stats[statevalf,cdf,normald]($Z1))");
$Pre2 = maple("(stats[statevalf,cdf,normald]($Z2))");
$P1=decimal(4,$Pre1);
$P2=decimal(4,$Pre2);
$Ans = decimal(4,$P2-$P1);
$PER = 100*$p;@
qu.4.37.uid=0361cf66-3946-4fc7-87b1-4c6454dc31a6@
qu.4.37.info=  Type=numeric;
  Course=202;
  Keyword=binomial;
  Keyword=continuity correction;
@

qu.4.38.mode=Multiple Choice@
qu.4.38.name=18. Canned Fish@
qu.4.38.comment=<p>Let X be the average weight of a can in the sample. We want P(X < $X)</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mi mathvariant='normal'>$X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$U</mi></mrow><mrow><mfrac><mi mathvariant='normal'>$S</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$N</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$Z</mi></mrow></mfenced></mrow></mstyle></math> <font size="3" face="Times New Roman">= $DecAns&nbsp; or $Ans%<br />
</font></p>@
qu.4.38.editing=useHTML@
qu.4.38.solution=@
qu.4.38.algorithm=$X = range(240,260);
$U = range(245,255);
$S = range(12,20);
$N = range(15,22);
$Z = decimal(4,($X-$U)/($S/sqrt($N)));
$P = maple("(stats[statevalf,cdf,normald])($Z)");
condition:le($P,0.95);
$Q=18;
$Fish=switch(rint(4),"salmon","tuna","sardines","mackerel");
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");
$DecAns = decimal(4,$P);
$Ans=decimal(2,100*$DecAns);
$Alt1=decimal(2,$Ans+range(0.5,0.9,0.01)*(100-$Ans));
$Alt2=decimal(2,range(0.5,0.9,0.01)*$Ans);
$Alt3=decimal(2,0.5*($Ans+$Alt1));
$Alt4=decimal(2,0.5*($Ans+$Alt2));@
qu.4.38.uid=25139142-f5b3-448e-83e1-984555671eb1@
qu.4.38.info=  Type=MC;
@
qu.4.38.question=<div title="UW Statistics Bank/Continuous Distributions/Normal Distribution Models/Q$Q"><img hspace="4" align="$Align" title="Can [IMG" alt="Can" src="__BASE_URI__CPD/NDM/FishCan$Which.gif" />Cans of $Fish have a nominal net weight of $X g. However, due to variation in the canning process, the actual net weight has an approximate normal distribution with a mean of $U g and a standard deviation of $S g. According to Consumer Affairs, a sample of $N tins should have less than a 5% chance that the mean weight is less than $X g. What is the actual probability that a sample of $N tins will have a mean weight less than $X g?</div>@
qu.4.38.answer=1@
qu.4.38.choice.1=$Ans%@
qu.4.38.choice.2=$Alt1%@
qu.4.38.choice.3=$Alt2%@
qu.4.38.choice.4=$Alt3%@
qu.4.38.choice.5=$Alt4%@
qu.4.38.fixed=@

qu.5.topic=PDF&CDF@

qu.5.1.mode=Inline@
qu.5.1.name=11. CDF to PDF@
qu.5.1.comment=<p>First notice that since Z's range is [1,+&infin;) , then Y's range is [0,+&infin;). Now:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><mi>Y</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>log</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><msup><mi>&ExponentialE;</mi><mrow><mi>y</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></msup></mrow></mstyle></math></p>
<p>So the pdf of Y is given by:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>f</mi><mrow><mi>Y</mi></mrow></msub><mfenced open='(' close=')' separators=','><mrow><mi>y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>y</mi></mrow></mfrac><mfenced open='[' close=']' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></msup></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></msup></mrow></mstyle></math></p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$Q=11;
$Y2Top=range(1,3,1);@
qu.5.1.uid=97359675-c466-4070-a4e7-8813f82a7fe0@
qu.5.1.info=  Course=230;
  Keyword=Test;
@
qu.5.1.weighting=1@
qu.5.1.numbering=alpha@
qu.5.1.part.1.name=sro_id_1@
qu.5.1.part.1.editing=useHTML@
qu.5.1.part.1.choice.5=None of the above.@
qu.5.1.part.1.fixed=4@
qu.5.1.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>log</mi><mfenced open='(' close=')' separators=','><mrow><mi>y</mi></mrow></mfenced></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mspace height='0.0ex' width='1.0em' depth='0.0ex' linebreak='nobreak'/><mo lspace='0.0em' rspace='0.0em'></mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>1</mn></mrow></mstyle></math>@
qu.5.1.part.1.question=null@
qu.5.1.part.1.choice.3=<em>-2log(y), y</em> > 0@
qu.5.1.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfrac><mrow><mi mathvariant='normal'>$Y2Top</mi></mrow><mrow><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'></mo><mspace height='0.0ex' width='1.0em' depth='0.0ex' linebreak='nobreak'/><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>1</mn></mrow></mstyle></math>@
qu.5.1.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&ExponentialE;</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></msup><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'></mo><mspace height='0.0ex' width='1.0em' depth='0.0ex' linebreak='nobreak'/><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn></mrow></mstyle></math>@
qu.5.1.part.1.mode=Multiple Choice@
qu.5.1.part.1.display=vertical@
qu.5.1.part.1.answer=1@
qu.5.1.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">Suppose Z has a cumulative distribution function given by:&nbsp;<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi>z</mi></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mspace height='0.0ex' width='1.0em' depth='0.0ex' linebreak='nobreak'/><mi>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>1</mn></mrow></mstyle></math></p><p>Let <em>Y = log(Z)</em>. Then the probability density of Y is:</p><p><span> </span><1><span> </span></p><p>&nbsp;</p></div>@

qu.5.2.mode=Multiple Choice@
qu.5.2.name=10. pdf of a function of a r.v.@
qu.5.2.comment=<p>First find the cumulative distribution function of Y:  <br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mi>y</mi></mrow></mfenced></mrow><mrow></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mi>P</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mi>y</mi></mrow></mfenced></mrow><mrow></mrow></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mn>5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>y</mi></mrow></msqrt></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>P</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>X</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&le;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><msup superscriptshift='0'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>5</mn><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mfrac></mrow></msup><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><munderover accent='false' accentunder='false'><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='prefix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='true' movablelimits='unset' accent='unset'>&Integral;</mo><mrow><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>&infin;</mi></mrow></mrow><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><mrow><msup superscriptshift='0'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>5</mn><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mfrac></mrow></msup><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi></mrow></mrow></munderover><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>f</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&InvisibleTimes;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>x</mi></mrow></mfenced><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&DifferentialD;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>x</mi></mrow></mrow><mrow></mrow></mrow></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><msub subscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>F</mi><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>X</mi></mrow></msub><mfenced><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><msup superscriptshift='0'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>5</mn><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mfrac></mrow></msup><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi></mrow></mfenced><mrow></mrow></mrow></mrow></math><br />
Now differentiate wrt y:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><msub subscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>f</mi><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>Y</mi></mrow></msub><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi></mrow></mfenced><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><msup superscriptshift='0'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>5</mn><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mfrac></mrow></msup></mrow></mrow><mrow><msub subscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>f</mi><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>X</mi></mrow></msub><mfenced><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><msup superscriptshift='0'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>5</mn><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mfrac></mrow></msup><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi></mrow></mfenced><mrow></mrow></mrow></mrow></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mfrac><mfrac><mn>3</mn><mn>4</mn></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mi>y</mi><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mfrac></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mi>y</mi><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mi>y</mi><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mfrac></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>y</mi><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow><mn>5</mn></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>for</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow><mrow></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mstyle></math></p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=@
qu.5.2.uid=306f28af-8561-43f5-a555-1ab065ba9eba@
qu.5.2.info=  Course=230;
  Type=MorithmicC;
  Algorithmic=no;
@
qu.5.2.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q10">Suppose a random variable X has a continuous distribution with probability density function <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>f</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>x</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>3</mn><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>4</mn></mfrac><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>x</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='true' lspace='0em' rspace='verythickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>x</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='true' lspace='0em' rspace='verythickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&comma;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>0</mn><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&le;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>x</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&le;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>2</mn></mrow></mrow></mrow></math>. Define a random variable <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mn>5</mn></mrow></msqrt><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced></mrow></mstyle></math>. Then the probability density function of Y is:</div>@
qu.5.2.answer=1@
qu.5.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>3</mn><mrow><mn>4</mn><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow><mn>5</mn></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>for</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msqrt><mrow><mn>5</mn></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mrow></mstyle></math>@
qu.5.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>@
qu.5.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt><mfenced open='(' close=')' separators=','><mrow><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow></mstyle></math>@
qu.5.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msup><mi>y</mi><mrow><mn>2</mn></mrow></msup></mrow><mn>5</mn></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced></mrow></mstyle></math>@
qu.5.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mi>y</mi><mrow><msqrt><mrow><mn>5</mn></mrow></msqrt></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.5.2.fixed=@

qu.5.3.mode=Inline@
qu.5.3.name=14. Find the pdf's graph@
qu.5.3.comment=<p>The trick here is that the function must be continuous at x = 0 . This means in turn that <font size="3" face="Times New Roman"><em>k(</em>$c<em>)<sup>2</sup></em> = 1</font>, or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$k</mi></mrow></mstyle></math> . Now differentiate the cdf to find the pdf:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'></mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$k2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$Q=14;
$lbot=range(1,5,1);
$ltop=range(1,$lbot,1);
$c=maple("simplify($ltop/$lbot)");
$k=maple("simplify($lbot^2/$ltop^2)");
$k2=maple("simplify(2*$k)");
$Plot1=plotmaple("plot(piecewise(x < -$c, 0, x<=0,$k2*(x+$c),x >=0, 0),x=-2..2),plotoptions	='width =350, height=200'");
$Plot2=plotmaple("plot(piecewise(x < -$c, 0, x<=0,$k*(x+$c),x >=0, 0),x=-2..2),plotoptions	='width =350, height=200'");
$Plot3=plotmaple("plot(piecewise(x < -$c, 0, x<=0,$k2*(x+$c),x >=0, 1),x=-2..2),plotoptions	='width =350, height=200'");
$Plot4=plotmaple("plot(piecewise(x < -$c, 1, x<=0,($k/3)*(x+$c)^3,x >=0, x),x=-2..2),plotoptions	='width =350, height=200'");@
qu.5.3.uid=20af1d9b-9e23-41aa-953d-3e0e42ef6c24@
qu.5.3.info=  Course=230;
  Type=MC;
@
qu.5.3.weighting=1@
qu.5.3.numbering=alpha@
qu.5.3.part.1.name=sro_id_1@
qu.5.3.part.1.editing=useHTML@
qu.5.3.part.1.choice.5=None of the above.<br>@
qu.5.3.part.1.fixed=4@
qu.5.3.part.1.choice.4=$Plot4<br>@
qu.5.3.part.1.question=null@
qu.5.3.part.1.choice.3=$Plot3<br>@
qu.5.3.part.1.choice.2=$Plot2<br>@
qu.5.3.part.1.choice.1=$Plot1<br>@
qu.5.3.part.1.mode=Multiple Choice@
qu.5.3.part.1.display=vertical@
qu.5.3.part.1.answer=1@
qu.5.3.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">A continuous random variable has its cumulative distribution function of the form:<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'></mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>0</mi></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>1</mi></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p><p>Which of the following is the graph of the pdf for this distribution? <1><span> </span></p></div>@

qu.5.4.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">Find the value of <em>k</em> (3 decimals or a proper fraction) that makes the following function a pdf:<br />
<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' mathbackground='#ffffff' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow><mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&amp;lcub;</mo><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>kx</mi><mfenced open='(' close=')' separators=','><mrow><mn>$n</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd></mtr></mtable><mo lspace='0.0em' rspace='0.0em'></mo></mrow><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>$n</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>otherwise</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mrow></mstyle></math></div>@
qu.5.4.answer.num=$kis@
qu.5.4.answer.units=@
qu.5.4.showUnits=false@
qu.5.4.grading=toler_abs@
qu.5.4.err=.005@
qu.5.4.negStyle=minus@
qu.5.4.numStyle=thousands scientific dollars arithmetic@
qu.5.4.mode=Numeric@
qu.5.4.name=09. Find k to make a pdf@
qu.5.4.comment=<p>To find <em>k</em>, integrate the given function over [-&infin;,+&infin;] (which means over [0,$n] since f(x) = 0 outside that interval), set the result equal to 1 and solve for k.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mi>&#8734;</mi></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&#8734;</mi></mrow></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi>kx</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>x</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>k</mi><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>k</mi></mrow><mrow><msubsup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>$n</mi><mrow><mn>3</mn></mrow></msup></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></mstyle></math></p>
<p>Set equal to 1 and solve for k:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>6</mn><mrow><msup><mi mathvariant='normal'>$n</mi><mrow><mn>3</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math> or $kis .</p>
<p>&nbsp;</p>@
qu.5.4.editing=useHTML@
qu.5.4.solution=@
qu.5.4.algorithm=$Q=9;
$n=range(1,8,1);
$kis = decimal(3,6/($n)^3);
$ktop=6;
$kbottom=$n^3;@
qu.5.4.uid=ef2239d4-1107-499b-993d-36ef4eb229da@
qu.5.4.info=  Course=230;
  Type=numeric;
@

qu.5.5.mode=True False@
qu.5.5.name=04. Is this a pdf?@
qu.5.5.comment=<p>Yes, it is a pdf for f(x) &ge; 0 everywhere and:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mi>&#8734;</mi></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&#8734;</mi></mrow></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></munderover><mfrac><mi mathvariant='normal'>$ktop</mi><mrow><mi mathvariant='normal'>$kbot</mi></mrow></mfrac><mi>x</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>x</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' mathbackground='#ffffff'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mfenced open='' close='&RightBracketingBar;' separators=','><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>$ktop</mn><mrow><mn>$kbot</mn></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>4</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><mn>8</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow></mrow></mstyle></math></p>
<p>That is, the function integrates to 1. Thus it is a pdf.</p>@
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qu.5.5.solution=@
qu.5.5.algorithm=$Q=4;
$ktop=3;
$n=range(2,8,1);
$kbot = 6*$n-8;@
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@
qu.5.5.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">The following function is a pdf:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mfrac><mrow><mi mathvariant='normal'>$ktop</mi></mrow><mrow><mi mathvariant='normal'>$kbot</mi></mrow></mfrac><mi>x</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd></mtr></mtable><mo lspace='0.0em' rspace='0.0em'></mo></mrow><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>otherwise</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mrow></mstyle></math> <br />
<p>&nbsp;</p>
</div>@
qu.5.5.answer=1@
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qu.5.5.fixed=@

qu.5.6.mode=Inline@
qu.5.6.name=18. Median given cdf II@
qu.5.6.comment=<p>The trick here is that the function must be continuous at x = 0 . This means in turn that <font size="3" face="Times New Roman"><em>k</em>($c)<sup>2</sup> = 1</font>, or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><font size="3" face="Times New Roman">$kML</font>.&nbsp; Now to be the median <font size="3" face="Times New Roman"><em>m</em></font> what the definition really means is <em><font size="3" face="Times New Roman">F</font></em><font size="3" face="Times New Roman">(</font><em><font size="3" face="Times New Roman">m</font></em><font size="3" face="Times New Roman">)</font><em><font size="3" face="Times New Roman"> = </font></em><font size="3" face="Times New Roman">0.5</font><em>. </em>Thus :</p>
<p><font size="3" face="Times New Roman">$kML(<em>m</em> + $cML)<sup>2</sup> =</font> <font size="3" face="Times New Roman">$OneOverTwokML</font><br />
<font size="3" face="Times New Roman">(<em>m</em> + $cML)<sup>2</sup> =</font> <font size="3" face="Times New Roman">$OneOverTwokML</font><br />
<font size="3" face="Times New Roman"><em>m</em> = -$cML</font> <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&plusmn;</mo></mrow><mrow><msqrt><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$OneOverTwok</mi></mrow></msqrt></mrow></mrow></mstyle></math></p>
<p>The definition of <font size="3" face="Times New Roman"><em>F</em>(<em>x</em>)</font> tells us to take the root&nbsp; <font size="3" face="Times New Roman">-$cML</font><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msqrt><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$OneOverTwok</mi></mrow></msqrt></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, so <br />
<font size="3" face="Times New Roman"><em>m</em> = $m</font></p>@
qu.5.6.editing=useHTML@
qu.5.6.solution=@
qu.5.6.algorithm=$Q=18;
$lbot=range(1,5);
$ltop=range(1,$lbot);
$c=maple("simplify($ltop/$lbot)");
$cML=mathml("$ltop/$lbot");
$k=maple("simplify($lbot^2/$ltop^2)");
$lbot2=$lbot^2;
$ltop2=$ltop^2;
$kML=mathml("$lbot2/$ltop2");
$Twok=2*$k;
$OneOverTwokML=mathml("$ltop2/(2*$lbot2)");
$OneOverTwok=$ltop2/(2*$lbot2);
$m=decimal(4,-$c+sqrt(1/(2*$k)));@
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qu.5.6.part.1.numStyle=thousands scientific  arithmetic@
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qu.5.6.part.1.showUnits=false@
qu.5.6.part.1.err=0.0010@
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qu.5.6.part.1.answer.num=$m@
qu.5.6.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">A continuous random variable has its cumulative distribution function of the form:<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'></mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p><p><span>Then the median of this distribution, to 4 decimals, is:<br /></span></p><p><1></p><p><span><em>(The median is the value m with the property that P(x</em></span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></mstyle></math>m) = P(x<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&ge;</mo></mrow></mstyle></math>m).)<span><br /></span></p></div>@

qu.5.7.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">Let X be a r.v. with a pdf f(x) defined as:<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mrow><mn>0</mn></mrow><mrow><mfrac><mrow><mi mathvariant='normal'>$nM1</mi></mrow><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup></mrow></mfrac></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac linethickness='0'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>1</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mn>1</mn></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.Find E(X). (4 decimal accuracy)</div>@
qu.5.7.answer.num=$nM1/$nM2@
qu.5.7.answer.units=@
qu.5.7.showUnits=false@
qu.5.7.grading=toler_abs@
qu.5.7.err=.0001@
qu.5.7.negStyle=minus@
qu.5.7.numStyle=thousands scientific dollars arithmetic@
qu.5.7.mode=Numeric@
qu.5.7.name=15. E[of a Continuous PDF]@
qu.5.7.comment=<p>Just integrate xf(x) from 1 to &infin;.&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>&#8734;</mi></mrow></munderover><mi mathvariant='normal'>xf</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>1</mn></mrow><mrow><mi>&infin;</mi></mrow></munderover><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mrow><mi mathvariant='normal'>$nM1</mi></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$nM1</mi><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>1</mn></mrow><mrow><mi>&infin;</mi></mrow></munderover><msup><mi>x</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$nM1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mfenced open='' close='|' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>$nM1</mi></mrow><mrow><mi>$nM2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mrow><mi>$nM2</mi></mrow></msup></mrow></mfrac></mrow></mrow></mfenced><mrow><mn>1</mn></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>$nM1</mi><mrow><mi>$nM2</mi></mrow></mfrac></mrow></mrow><mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.5.7.editing=useHTML@
qu.5.7.solution=@
qu.5.7.algorithm=$Q=15;
$n=range(3,8,1);
$nM1=$n-1;
$nM2=$n-2;@
qu.5.7.uid=c4775d66-60e7-47d8-b72b-9303a15d61e9@
qu.5.7.info=  Course=230;
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@

qu.5.8.mode=Inline@
qu.5.8.name=13. Find k and then pdf@
qu.5.8.comment=<p>The trick here is that the function must be continuous at x = 0 . This means in turn that <em>k($c)<sup>2</sup> = 1</em>, or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><msup><mi mathvariant='normal'>$c</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$k</mi></mrow></mstyle></math> . Now differentiate the cdf to find the pdf:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac></mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'></mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$k2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.5.8.editing=useHTML@
qu.5.8.solution=@
qu.5.8.algorithm=$Q=13;
$lbot=range(1,5,1);
$ltop=range(1,$lbot,1);
$c=maple("simplify($ltop/$lbot)");
$k=maple("simplify($lbot^2/$ltop^2)");
$k2=maple("simplify(2*$k)");@
qu.5.8.uid=cb1e48c0-28c8-4427-a803-7dfa0881634f@
qu.5.8.info=  Course=230;
  Keyword=Test;
  Type=numeric;
@
qu.5.8.weighting=1@
qu.5.8.numbering=alpha@
qu.5.8.part.1.comment.3=@
qu.5.8.part.1.comment.2=@
qu.5.8.part.1.name=sro_id_1@
qu.5.8.part.1.comment.1=@
qu.5.8.part.1.editing=useHTML@
qu.5.8.part.1.choice.5=None of the above.@
qu.5.8.part.1.fixed=4@
qu.5.8.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'></mo><mi mathvariant='normal'>$k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mrow><mn>3</mn></mrow></msup></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>x</mi></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math>@
qu.5.8.part.1.question=null@
qu.5.8.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>0</mi></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$k2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math>@
qu.5.8.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>0</mi></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math>@
qu.5.8.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>0</mi></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi mathvariant='normal'>$k2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math>@
qu.5.8.part.1.mode=Multiple Choice@
qu.5.8.part.1.display=vertical@
qu.5.8.part.1.comment.5=@
qu.5.8.part.1.comment.4=@
qu.5.8.part.1.answer=1@
qu.5.8.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">A continuous random variable has its cumulative distribution function of the form:&nbsp;<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'></mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p><p>Which of the following is the pdf for this distribution? <1><span> </span></p></div>@

qu.5.9.mode=Inline@
qu.5.9.name=17. Median given CDF@
qu.5.9.comment=<p>The median is simply that value <font size="3" face="Times New Roman"><em>z</em></font> with the property that <font size="3" face="Times New Roman">P(<em>Z</em> &le; <em>z</em>) = P(<em>Z</em> &ge; <em>z</em>)</font> .&nbsp; This just means the point <font size="3" face="Times New Roman"><em>z</em></font> where the CDF <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>z</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mstyle></math>.</p>
<p>Since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi>z</mi></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;<em> <font size="3" face="Times New Roman">z</font></em><font size="3" face="Times New Roman"> = 2</font> is the median.</p>@
qu.5.9.editing=useHTML@
qu.5.9.solution=@
qu.5.9.algorithm=$Q=17;
$AltTop=range(3,9,2);
$AltBot=range(2,$AltTop-1,2);
$AltInt= switch(rint(4),1,3,7,12);
$Alt1ML=mathml("$AltInt");
$Alt2ML=mathml("$AltTop/$AltBot");
$AltLog=range(2,7,1);@
qu.5.9.uid=99cd63b8-38e5-4761-bccf-ba143aed1c92@
qu.5.9.info=  Course=230;
  Type=MC;
@
qu.5.9.weighting=1@
qu.5.9.numbering=alpha@
qu.5.9.part.1.comment.3=@
qu.5.9.part.1.comment.2=@
qu.5.9.part.1.name=sro_id_1@
qu.5.9.part.1.comment.1=@
qu.5.9.part.1.editing=useHTML@
qu.5.9.part.1.choice.5=There is no median value for this distribution.@
qu.5.9.part.1.fixed=4@
qu.5.9.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi mathvariant='normal'>log</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$AltLog</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>@
qu.5.9.part.1.question=null@
qu.5.9.part.1.choice.3=<font size="3" face="Times New Roman">2</font>@
qu.5.9.part.1.choice.2=$Alt2ML@
qu.5.9.part.1.choice.1=$Alt1ML@
qu.5.9.part.1.mode=Multiple Choice@
qu.5.9.part.1.display=vertical@
qu.5.9.part.1.comment.5=@
qu.5.9.part.1.comment.4=@
qu.5.9.part.1.answer=3@
qu.5.9.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">Suppose Z has a cumulative distribution function given by:<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi>z</mi></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mspace height='0.0ex' width='1.0em' depth='0.0ex' linebreak='nobreak'/><mi>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>1</mn></mrow></mstyle></math></p><p>Then the median value of Z is: <br />(The <u>median value</u> is the point <font size="3" face="Times New Roman"><em>z = m</em></font> with the property that  <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>m</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>m</mi></mrow></mfenced></mrow></mstyle></math>)</p><p><span> </span><1><span> </span></p></div>@

qu.5.10.mode=Inline@
qu.5.10.name=12. Which graph?@
qu.5.10.comment=<p>One of the properties of a cdf is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi mathcolor='#800080'>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi>&infin;</mi></mrow></munder><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mn>1</mn></mrow></mstyle></math>. This eliminates these two graphs:<br />
$Plot3 <br />
$Plot4</p>
<p>The only points of possible non-differentiability are <em>x = -1</em> and <em>x = 0</em>. That eliminates the following graph, which has such a point at <em>x = -1/2</em>.</p>
<p>$Plot2</p>
<p>Finally to be a cdf we need <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi mathcolor='#800080'>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mn>0</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi mathcolor='#800080'>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mn>0</mn><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo></mrow></msup></mrow></munder><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> (continuity) which implies <em>k = 1</em>. The remaining graph then is the correct one:</p>
<p>$Plot1</p>@
qu.5.10.editing=useHTML@
qu.5.10.solution=@
qu.5.10.algorithm=$Q=12;
$Plot1=plotmaple("plot(piecewise(x < -1, 0, x<=0,(1+x)^2,x >= 0, 1),x=-2..2),plotoptions	='width	=350, height=200'");
$Plot2=plotmaple("plot(piecewise(x < -1, 0, x<=-0.5,4*(1+x)^2,x >= -0.5, 1),x=-2..2),plotoptions	='width	=350, height=200'");
$Plot3=plotmaple("plot(piecewise(x < -1, 0, x<=0,(1+x)^2,x < 1, (x-1)^2,x>=1,0),x=-2..2),plotoptions	='width	=350, height=200'");
$Plot4=plotmaple("plot(piecewise(x < -1, 0, x<=0,2*(1+x)^2,x >= 0, 2),x=-2..2),plotoptions	='width	=350, height=200'");@
qu.5.10.uid=a6865faf-99af-4b23-8a4e-6dc80f3f4f5f@
qu.5.10.info=  Course=230;
  Keyword=Test;
  Type=MC;
  Algorithmic=no;
@
qu.5.10.weighting=1@
qu.5.10.numbering=alpha@
qu.5.10.part.1.name=sro_id_1@
qu.5.10.part.1.editing=useHTML@
qu.5.10.part.1.choice.5=None of the above.<br>@
qu.5.10.part.1.fixed=4@
qu.5.10.part.1.choice.4=$Plot4<br>@
qu.5.10.part.1.question=null@
qu.5.10.part.1.choice.3=$Plot3<br>@
qu.5.10.part.1.choice.2=$Plot2<br>@
qu.5.10.part.1.choice.1=$Plot1<br>@
qu.5.10.part.1.mode=Multiple Choice@
qu.5.10.part.1.display=vertical@
qu.5.10.part.1.answer=1@
qu.5.10.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">A continuous random variable has its cumulative distribution function of the form: <br /><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F(x)=</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>0</mi></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>x</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p><p>Which of the following is  the graph of  this cdf?</p><p><1><span> </span></p></div>@

qu.5.11.mode=True False@
qu.5.11.name=03. Is this a pdf?@
qu.5.11.comment=<p>No, it is not a pdf. It's true that f(x) &ge; 0 everywhere but:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mi>&#8734;</mi></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&#8734;</mi></mrow></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></munderover><mfrac><mrow><mi mathvariant='normal'>$ktop</mi></mrow><mrow><mi mathvariant='normal'>$kbot</mi></mrow></mfrac><mi>x</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mn>x</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mfenced open='' close='&RightBracketingBar;' separators=','><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$ktop</mi></mrow><mrow><mi mathvariant='normal'>$kbot</mi></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mrow><mfrac><mi mathvariant='normal'>$n</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mfrac><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$ktop</mi></mrow><mrow><mi mathvariant='normal'>$kbot</mi></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>8</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$ktop</mi><mrow><mi mathvariant='normal'>$kbot</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>8</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$ktop</mi><mrow><mi mathvariant='normal'>$kbot</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mi mathvariant='normal'>$CalcTop</mi><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi mathvariant='normal'>$CalcTop</mi><mrow><mi mathvariant='normal'>$kbot</mi></mrow></mfrac></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p>
<p>That is, the function does not integrate to 1. Thus it is NOT a pdf.</p>@
qu.5.11.editing=useHTML@
qu.5.11.solution=@
qu.5.11.algorithm=$Q=3;
$ktop=3;
$n=range(2,8,1);
$n2=2*$n;
$kbot = 8*$n-6;
$CalcTop = 6*$n-8;@
qu.5.11.uid=5835c980-42ee-4e80-9803-e353e23a9d6e@
qu.5.11.info=  Course=230;
  Type=TF;
@
qu.5.11.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">The following function is a pdf:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mfrac><mrow><mi mathvariant='normal'>$ktop</mi></mrow><mrow><mi mathvariant='normal'>$kbot</mi></mrow></mfrac><mi>x</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd></mtr></mtable><mo lspace='0.0em' rspace='0.0em'></mo></mrow><mrow><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>otherwise</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mrow></mstyle></math></div>@
qu.5.11.answer=2@
qu.5.11.choice.1=True@
qu.5.11.choice.2=False@
qu.5.11.fixed=@

qu.5.12.mode=Multiple Choice@
qu.5.12.name=05. Find alpha to make a pdf.@
qu.5.12.comment=<p>Just integrate the function, set the result to 1 and solve for &alpha;: <br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&#945;</mi><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&#8722;</mo><mrow><mfrac><mi>&#960;</mi><mn>2</mn></mfrac></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mi>&#960;</mi><mn>2</mn></mfrac></mrow></mrow></munderover><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mfenced open='' close='&RightBracketingBar;' separators=','><mrow><mfrac><mrow><mi>&alpha;</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>2</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>2</mn></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow></mrow><mrow><mfrac><mi>&alpha;</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>So</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mi>&#945;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$n</mi><mn>2</mn></mfrac><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mrow></mstyle></math></p>@
qu.5.12.editing=useHTML@
qu.5.12.solution=@
qu.5.12.algorithm=$Q=5;
$n=range(1,5,1);
$nx = $n*x;
$ans = $n/2;
$alt1 = $ans-1.5;@
qu.5.12.uid=57be80e0-a669-4867-8ab9-13d20a4e7eee@
qu.5.12.info=  Course=230;
  Type=MC;
@
qu.5.12.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">The probability density function of a r.v. X is given by: <br />
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mi mathvariant='normal'>x</mi></mrow></mfenced></mrow><mrow><mn>0</mn></mrow></mfrac><mfrac linethickness='0'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>2</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>&pi;</mi><mo lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&sol;</mo><mn>2</mn></mrow><mi mathvariant='normal'>otherwise</mi></mfrac><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> .&nbsp;</p>
<p>Then &alpha; is:</p>
</div>@
qu.5.12.answer=3@
qu.5.12.choice.1=0.1@
qu.5.12.choice.2=$alt1@
qu.5.12.choice.3=$ans@
qu.5.12.choice.4=2/&#960;@
qu.5.12.choice.5=Cannot be determined with this information.@
qu.5.12.fixed=4@

qu.5.13.mode=Multiple Choice@
qu.5.13.name=01. Given pdf, find cdf@
qu.5.13.comment=<p>Let F(x) be the cdf.</p>
<ul>
    <li>Then F(x) = 0 for all x < 0, since f(x) is.&nbsp;</li>
    <li>F(x) = 1 for x > 2, since, by the properties of a cdf.</li>
    <li>For x between 0 and 2 just integrate the pdf:<br />
    <br />
    <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi>x</mi></mrow></munderover><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$kt</mi><mrow><mi mathvariant='normal'>$kb</mi></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mi>x</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mi>t</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>t</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$kt</mi><mrow><mi mathvariant='normal'>$kb</mi></mrow></mfrac><msubsup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$n</mi><msup><mi>t</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi>t</mi><mrow><mn>3</mn></mrow></msup><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mi>x</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$kt</mi><mrow><mi mathvariant='normal'>$kb</mi></mrow></mfrac><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$n</mi><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>x</mi><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></li>
</ul>@
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$n	=	range(2,8,1);
$kt	=	3;
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$k	=	$kt/$kb;@
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qu.5.13.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mrow><mfrac><mrow><mi mathvariant='normal'>$kt</mi></mrow><mrow><mi mathvariant='normal'>$kb</mi></mrow></mfrac><mi>x</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mn>0</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.0em' rspace='0.0em'></mo><mfrac linethickness='0'><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.3em' rspace='0.3em'></mo><mo lspace='10.3em' rspace='10.3em'></mo><mi mathvariant='normal'>otherwise</mi></mrow></mfrac></mrow></mstyle></math> is the pdf for a continuous r.v. X, what is the cdf for X?</div>@
qu.5.13.answer=1@
qu.5.13.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F(x)=</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mrow><mfrac><mi mathvariant='normal'>$kt</mi><mrow><mi mathvariant='normal'>$kb</mi></mrow></mfrac></mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$n</mi><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>x</mi><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced></mrow></mtd><mtd><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>2</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math>@
qu.5.13.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F(x)=</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mn>2</mn><mrow><mi mathvariant='normal'>$kb</mi></mrow></mfrac><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$n</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>x</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mtd><mtd><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>2</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math>@
qu.5.13.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F(x)=</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'></mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup></mrow></mfenced></mrow></mtd><mtd><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>2</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math>@
qu.5.13.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F(x)=</mi></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mi>x</mi><mrow><mi mathvariant='normal'>$kb</mi></mrow></mfrac></mrow></mtd><mtd><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>2</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math>@
qu.5.13.choice.5=The cdf is not shown here.@
qu.5.13.fixed=4@

qu.5.14.mode=Multiple Choice@
qu.5.14.name=07. Find α to make a cdf@
qu.5.14.comment=To remind you:<br><br>A fuunction F is a cdf iff it has the following properties:<ol><li><i>F</i> is non-decreasing.</li><li><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><munder accentunder='false'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>lim</mi><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='postfix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='true'>&rarr;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo></mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>&infin;</mi></mrow></mrow></munder><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>F</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>0</mn><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='true' lspace='0em' rspace='verythickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='true' lspace='0em' rspace='verythickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&comma;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo></mrow><mrow><munder accentunder='false'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>lim</mi><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='postfix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='true'>&rarr;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>&infin;</mi></mrow></mrow></mrow></munder><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>F</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn></mrow></mrow></math></li><li>F is right-continuous<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><munder accentunder='false'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>lim</mi><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='postfix' fence='unset' separator='unset' lspace='0em' rspace='0em' stretchy='true' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='true'>&rarr;</mo><msup superscriptshift='0'><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>a</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='mediummathspace' rspace='mediummathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&plus;</mo></msup></mrow></mrow></munder></mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>F</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>y</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>F</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&ApplyFunction;</mo><mfenced><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>a</mi></mrow></mfenced></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&forall;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>a</mi></mrow></mrow></math></li></ol>
<ol><li>
We must have &#945; &#8805; 0 (otherwise F(x) is decreasing for 0 < x &#8804; 2, which does not meet the criteria for a pdf).
<li>Regardless of the value of &#945; notice that these limit conditions are met.
<li>If &#945; > 0 then this condition is met everywhere (you need only check at x = 0 and x = 2).
      What if &#945; = 0? Then F(x) is NOT right-continuous at x = 0.</li></ol>

Conclusion: To make F(x) a pdf we must have &#945; > 0.@
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@
qu.5.14.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q07">For what value of &alpha; is:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x &#8804;</mi><mn> 0</mn></mrow></mtd></mtr><mtr><mtd><mrow><msup><mi>x</mi><mrow><mi>&alpha;</mi></mrow></msup></mrow></mtd><mtd><mrow><mn>0</mn><mi> &amp;lt; x &#8804;</mi><mn> 1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mspace height='0.0ex' width='1.0em' depth='0.0ex' linebreak='nobreak'/><mi>&gt;</mi><mn>1</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math><br />
a pdf?</div>@
qu.5.14.answer=1@
qu.5.14.choice.1=&alpha; > 0@
qu.5.14.choice.2=&alpha; < 0@
qu.5.14.choice.3=&alpha; = 1@
qu.5.14.choice.4=0 < &alpha; < 1@
qu.5.14.choice.5=No value of &alpha; can make this function a cdf.@
qu.5.14.fixed=4@

qu.5.15.mode=Inline@
qu.5.15.name=19. Find k to make a CDF@
qu.5.15.comment=<p>The trick here is that the function must be continuous at <font size="3" face="Times New Roman"><em>x</em> = 0</font> . This means in turn that <font size="3" face="Times New Roman"><em>k</em>($c)<sup>2</sup> = 1</font>, or <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$c</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$k</mi></mrow></mstyle></math></p>@
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qu.5.15.solution=@
qu.5.15.algorithm=$Q=19;
$lbot=range(2,7);
$ltop=range(1,$lbot-1);
$c=maple("simplify($ltop/$lbot)");
$lbot2=$lbot^2;
$ltop2=$ltop^2;
$k=maple("simplify($lbot^2/$ltop^2)");@
qu.5.15.uid=0c1f6866-daf3-4b21-9856-52c3abba796c@
qu.5.15.info=  Course=230;
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qu.5.15.numbering=alpha@
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qu.5.15.part.1.answer.num=$k@
qu.5.15.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">A continuous random variable has its cumulative distribution function of the form:&nbsp;<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'></mo><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>0</mi></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mtd><mtd><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>1</mi></mrow></mtd><mtd><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p><p>Then k is (3 decimals, or as a fraction): <span> </span><1><span> </span></p></div>@

qu.5.16.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">Suppose <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>y</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mrow><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>y</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup></mrow><mrow><mn>0</mn></mrow></mfrac><mfrac linethickness='0'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>2</mn></mrow><mrow><mi mathvariant='normal'>otherwise</mi></mrow></mfrac></mrow></mstyle></math>is a pdf for a continuous random variable <em>Y</em> .
<p>Find P(Y &le; 1) (3 decimal accuracy).</p>
<p>&nbsp;</p>
</div>@
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qu.5.16.mode=Numeric@
qu.5.16.name=08. Find P(Y<=1) from a pdf@
qu.5.16.comment=<p>First: use the definition of a pdf to find k by integrating the function and setting the integral to 1:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>k</mi><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></munderover><msup><mi>y</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mfenced open='' close='&RightBracketingBar;' separators=','><mrow><mfrac><mi>k</mi><mrow><mi mathvariant='normal'>$np1</mi></mrow></mfrac><msup><mi>y</mi><mrow><mi mathvariant='normal'>$np1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mn>2</mn><mrow><mi mathvariant='normal'>$np1</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$np1</mi></mrow></mfrac></mrow><mrow></mrow></mstyle></math> = 1</p>
<p>So <em>k</em> = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$np1</mi><mrow><msup><mn>2</mn><mrow><mi mathvariant='normal'>$np1</mi></mrow></msup></mrow></mfrac></mrow></mstyle></math> . Then find P(Y &le; 1) by integrating the pdf from 0 to 1 (using the <em>k</em> you just found):</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$np1</mi><mrow><msup><mn>2</mn><mrow><mi mathvariant='normal'>$np1</mi></mrow></msup></mrow></mfrac><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></munderover><msup><mi>y</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mfenced open='' close='&RightBracketingBar;' separators=','><mrow><mfrac><mi mathvariant='normal'>$np1</mi><mrow><msup><mn>2</mn><mrow><mi mathvariant='normal'>$np1</mi></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><msup><mi>y</mi><mrow><mi mathvariant='normal'>$np1</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$np1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><msup><mn>2</mn><mrow><mi mathvariant='normal'>$np1</mi></mrow></msup></mrow></mfrac></mrow></mstyle></math>.</p>@
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qu.5.16.algorithm=$Q=8;
$n=range(2,8);
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qu.5.17.mode=Inline@
qu.5.17.name=16. CDF to PDF, then E() which DNE@
qu.5.17.comment=<p>To find Expected Value you need the pdf. To find this differentiate the cdf:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mn>1</mn><mrow><mi>z</mi></mrow></mfrac></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'></mo><mspace height='0.0ex' width='1.0em' depth='0.0ex' linebreak='nobreak'/><mi>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>1</mn></mrow></mstyle></math></p>
<p>Then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>E</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mrow><mi>&infin;</mi></mrow></mrow></munderover><mi>z</mi><mi mathcolor='#0000ff'>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>z</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>1</mn></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>&infin;</mi></mrow></munderover><mi>z</mi><mfrac><mn>1</mn><mrow><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>z</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>z</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&RightBracketingBar;</mo><msubsup><mi></mi><mrow><mn>1</mn></mrow><mrow><mi>&infin;</mi></mrow></msubsup></mrow><mrow></mrow></mstyle></math> which is undefined. Thus the Expected Value does not exist.</p>@
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qu.5.17.part.1.comment.1=@
qu.5.17.part.1.name=sro_id_1@
qu.5.17.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q$Q">Suppose Z has a cumulative distribution function given by:<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi>z</mi></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mspace height='0.0ex' width='1.0em' depth='0.0ex' linebreak='nobreak'/><mi>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>1</mn></mrow></mstyle></math></p><p>Then E(Z) is:</p><p><span> </span><1><span> </span></p></div>@

qu.5.18.mode=Multiple Choice@
qu.5.18.name=06. Find α to make a cdf@
qu.5.18.comment=@
qu.5.18.editing=useHTML@
qu.5.18.solution=@
qu.5.18.algorithm=@
qu.5.18.uid=13dd8e0b-cf68-44b8-a905-4f072733bc30@
qu.5.18.info=  Course=230;
  Type=MC;
  Algorithmic=no;
@
qu.5.18.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q06">For what value of <font size="3" face="Times New Roman"><em>&alpha;</em></font> is:<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfenced open='' close='' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd><mtd><mrow><mi>x &#8804;</mi><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><msup><mi>x</mi><mrow><mi>&alpha;</mi></mrow></msup></mrow></mtd><mtd><mrow><mn>0</mn><mi>&amp;lt; x &#8804; </mi><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1</mn></mrow></mtd><mtd><mrow><mi>x</mi><mspace height='0.0ex' width='1.0em' depth='0.0ex' linebreak='nobreak'/><mi>&gt;</mi><mn>2</mn></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math><br />
a pdf?</div>@
qu.5.18.answer=5@
qu.5.18.choice.1=&alpha; > 0@
qu.5.18.choice.2=&alpha; < 0@
qu.5.18.choice.3=&alpha; = 1@
qu.5.18.choice.4=0 < &alpha; < 1@
qu.5.18.choice.5=This function can never be a cdf.@
qu.5.18.fixed=4@

qu.5.19.mode=Inline@
qu.5.19.name=02. Which is untrue?@
qu.5.19.comment=<p>By the definition of pdf &amp; cdf we have the following limit properties:</p>
<ol>
    <li><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi>&infin;</mi></mrow></munder></mrow><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'></mo><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi>&infin;</mi></mrow></munder><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math></li>
    <li><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math></li>
</ol>
<p>Property 1. tells us that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi>&infin;</mi></mrow></munder><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>is always untrue.</p>@
qu.5.19.editing=useHTML@
qu.5.19.solution=@
qu.5.19.algorithm=@
qu.5.19.uid=e2d814ea-4ce8-40a0-a6b0-0498774d7b70@
qu.5.19.info=  Course=230;
  Type=MC;
  Algorithmic=no;
@
qu.5.19.weighting=1@
qu.5.19.numbering=alpha@
qu.5.19.part.1.name=sro_id_1@
qu.5.19.part.1.editing=useHTML@
qu.5.19.part.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mrow><mi mathvariant='normal'>lim</mi></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi>&infin;</mi></mrow></munder><mi>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math><br>@
qu.5.19.part.1.fixed=@
qu.5.19.part.1.choice.4=If a > b then F(a) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&ge;</mo></mrow></mstyle></math> F(b)@
qu.5.19.part.1.question=null@
qu.5.19.part.1.choice.3=<i>f</i>(<i>x</i>) > 1 for some x.@
qu.5.19.part.1.choice.2=P(X <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&ge;</mo></mrow></mstyle></math> x) = P(X > x) for all x.@
qu.5.19.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi>&infin;</mi></mrow></munder><mi>F</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math><br>@
qu.5.19.part.1.mode=Non Permuting Multiple Choice@
qu.5.19.part.1.display=vertical@
qu.5.19.part.1.answer=1@
qu.5.19.question=<div title="UW Statistics Bank/Continuous Probability Models/PDF&amp;CDF/Q02">Suppose <font size="3" face="Times New Roman"><em>f</em>(<em>x</em>)</font> is a continuous pdf with a cdf of <font size="3" face="Times New Roman"><em>F</em>(<em>x</em>)</font>. Which of the following is ALWAYS untrue?<br /><br /><p><span> </span><1><span> </span></p></div>@

qu.6.topic=Basics@

qu.6.1.mode=Multiple Choice@
qu.6.1.name=01+. Which is continuous?@
qu.6.1.comment=<p>Only "$Ans" involves a continuous measurement, the rest are discrete.</p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$Q="01+";
$Alt1=switch(rint(2),"the number of brothers a randomly chosen person has","the number of birds swimming in a pond at a given moment");
$Alt2=switch(rint(2),"the number of cars owned by a randomly chosen adult male","the number of children living in a randomly chosen house");
$Ans=switch(rint(3),"the time it takes for a randomly chosen woman to run 100 meters","the volume of water flowing over Niagara Falls during a randomly chosen minute","the time it takes a randomly chosen person to commute to work");
$Alt3="number of orders received by a mail order company in a randomly chosen week";@
qu.6.1.uid=f98d95d4-ac2c-48fb-8c13-dda04c88e6c4@
qu.6.1.info=  Type=MC;
@
qu.6.1.question=<div title="UW Statistics Bank/Continuous Distributions/Basics/Q$Q">Which of the following random variables would be considered continuous?</div>@
qu.6.1.answer=2@
qu.6.1.choice.1=$Alt1@
qu.6.1.choice.2=$Ans@
qu.6.1.choice.3=$Alt2@
qu.6.1.choice.4=$Alt3@
qu.6.1.fixed=@

qu.6.2.mode=True False@
qu.6.2.name=03. Rationals & Irrationals@
qu.6.2.comment=The set of Rational numbers Q is countable, so 



<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>X</mi><mi>&#949;Q</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><munder><mrow><mi>&#931;</mi></mrow><mrow><mi>x&#949;Q</mi></mrow></munder></mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'></mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow>
</mstyle></math> since for a continuous distribution, the probability of any particular value P(X = x) is zero. Therefore, P(X &#8712; <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow>
<mrow><mover accent='false'><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>Q</mi></mrow><mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&minus;</mo></mrow></mover></mrow></mrow></math>) = 1 &#8722; P(X &#8712; Q) = 1 and the statement is false.@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=@
qu.6.2.uid=86fbe2c6-b92a-47c2-8011-e3fa6c16137b@
qu.6.2.info=  Course=230;
  Type=T/F;
  Algorithmic=no;
@
qu.6.2.question=<div title="UW Statistics Bank/Continuous Distributions/Basics/Q03">If X is a random variable with a continuous distribution, and Q is the set of rational numbers, so its complement <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Q</mi></mrow><mi>&minus;</mi></mover></mrow></mstyle></math> is the set of irrational numbers, then P(X&isin;Q) > P(X&isin;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Q</mi></mrow><mi>&minus;</mi></mover></mrow></mstyle></math>).</div>@
qu.6.2.answer=2@
qu.6.2.choice.1=True@
qu.6.2.choice.2=False@
qu.6.2.fixed=@

qu.6.3.mode=True False@
qu.6.3.name=02. Continuous pdf at a point@
qu.6.3.comment=The statement is true because for a continuous pdf <i>f(x)</i> we have <i>f(X=x) = 0</i> for <u>any</u> point in the domain of <i>f</i>.@
qu.6.3.editing=useHTML@
qu.6.3.solution=@
qu.6.3.algorithm=@
qu.6.3.uid=a80fb9bc-428f-41d0-9a20-56fbde1db3c8@
qu.6.3.info=  Course=230;
  Type=T/F;
  Algorithmic=no;
@
qu.6.3.question=<div title="UW Statistics Bank/Continuous Distributions/Basics/Q02">If X is a random variable with a continuous distribution, then for any values x and y, P(X = x) = P(X = y).</div>@
qu.6.3.answer=1@
qu.6.3.choice.1=True@
qu.6.3.choice.2=False@
qu.6.3.fixed=@

qu.6.4.mode=True False@
qu.6.4.name=04. P(X < x) vs P(X ≤ x)@
qu.6.4.comment=<p>False. <font size="3" face="Times New Roman"><em>P</em>(<em>X</em> &le; <em>x</em>) = <em>P</em>(<em>X</em> < <em>x</em>) + <em>P</em>(<em>X</em> = <em>x</em>) = <em>P</em>(<em>X</em> < <em>x</em>) + 0</font> since any continuous pdf is zero at any point.</p>@
qu.6.4.editing=useHTML@
qu.6.4.solution=@
qu.6.4.algorithm=@
qu.6.4.uid=32e40871-bdb4-4b29-8dd2-649cabdd2c48@
qu.6.4.info=  Author=Sean Scott;
  Type=T/F;
@
qu.6.4.question=<div title="UW Statistics Bank/Continuous Distributions/Basics/Q04">Let X be a continuous random variable defined on <strong>R</strong> . Then for any real number x:
<p>P(X < x) < P(X &le; x)</p>
</div>@
qu.6.4.answer=2@
qu.6.4.choice.1=True@
qu.6.4.choice.2=False@
qu.6.4.fixed=@

qu.7.topic=Uniform@

qu.7.1.question=<div title="UW Statistics Bank/Continuous Distributions/Uniform Distributions/Q$Q">If X is a random variable with the (continuous) distribution shown below, find (4 decimal accuracy) P(X > $BreakPt):
<p align="center"><applet width="443" height="165" code="applets.labelImage.LabelImage">
<param value="__BASE_URI__CPD/UD/Uniform.gif" name="image" />
<param value="1" name="size" />
<param value="33" name="label.1.x" />
<param value="67" name="label.1.y" />
<param value="$p" name="label.1.text" /> </applet></p>
</div>@
qu.7.1.answer.num=$Ans@
qu.7.1.answer.units=@
qu.7.1.showUnits=false@
qu.7.1.grading=toler_abs@
qu.7.1.err=.001@
qu.7.1.negStyle=minus@
qu.7.1.numStyle=thousands scientific dollars arithmetic@
qu.7.1.mode=Numeric@
qu.7.1.name=02. Uniform, find P(X>n)@
qu.7.1.comment=<p>Notice that the pdf is simply:<br />
<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mi mathvariant='normal'>$p</mi><mn>0</mn></mfrac><mrow><mfrac linethickness='0'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>&alpha;</mi></mrow><mi mathvariant='normal'>otherwise</mi></mfrac></mrow></mrow></mrow></mstyle></math><br />
<br />
So to find &alpha; you must integrate this over [0,&alpha;], set that integral equal to 1 and solve to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>&#945;</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$p</mi></mfrac></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>or</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$alpha</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&period;</mo></mrow></math><br />
Now <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi mathvariant='normal'>$BreakPt</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>1 -</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mi mathvariant='normal'>$BreakPt</mi></munderover><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$BreakPt</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=$Q=2;
$p=decimal(1,range(0.2,0.5,0.05));
$alpha=decimal(3,1/$p);
$BreakPt = range(1,int($alpha),1);
$Ans=decimal(4,1-$p*$BreakPt);@
qu.7.1.uid=fa035569-4368-416d-8341-14ef12264e98@
qu.7.1.info=  Course=230;
  Type=numeric;
@

qu.7.2.question=<div title="UW Statistics Bank/Continuous Distributions/Uniform Distributions/Q$Q">If X is a random variable with the (continuous) distribution shown below, find&nbsp; P(X &le; $BreakPt) (4 decimal accuracy).
<p align="center"><applet width="420" height="163" code="applets.labelImage.LabelImage">
<param value="__BASE_URI__CPD/UD/Uniform.gif" name="image" />
<param value="2" name="size" />
<param value="12" name="label.1.x" />
<param value="61" name="label.1.y" />
<param value="$p" name="label.1.text" />
<param value="326" name="label.2.x" />
<param value="49" name="label.2.y" />
<param value="$p" name="label.2.text" /></applet></p>
</div>@
qu.7.2.answer.num=$Ans@
qu.7.2.answer.units=@
qu.7.2.showUnits=false@
qu.7.2.grading=toler_abs@
qu.7.2.err=0.001@
qu.7.2.negStyle=minus@
qu.7.2.numStyle=thousands scientific dollars arithmetic@
qu.7.2.mode=Numeric@
qu.7.2.name=01. Uniform, find P(X≤n)@
qu.7.2.comment=<p>Notice that the pdf is simply:<br />
<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mi mathvariant='normal'>$p</mi><mn>0</mn></mfrac><mrow><mfrac linethickness='0'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>&alpha;</mi></mrow><mi mathvariant='normal'>otherwise</mi></mfrac></mrow></mrow></mrow></mstyle></math><br />
<br />
So to find &alpha; you must integrate this over <font size="3" face="Times New Roman">[0,<em>&alpha;</em>]</font>, set that integral equal to 1 and solve to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>&#945;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mi mathvariant='normal'>$p</mi></mfrac></mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>or</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>$alpha</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math><br />
Now <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#8804;</mo><mi mathvariant='normal'>$BreakPt</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mn>0</mn></mrow><mi mathvariant='normal'>$BreakPt</mi></munderover><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&#8901;</mo><mi mathvariant='normal'>$BreakPt</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=$Q=1;
$p=decimal(1,range(0.2,0.5,0.05));
$alpha=decimal(3,1/$p);
$BreakPt = range(1,int($alpha),1);
$Ans=decimal(4,$p*$BreakPt);@
qu.7.2.uid=f6b73320-7899-47eb-ab26-d654865169b1@
qu.7.2.info=  Course=230;
  Type=numeric;
@

qu.7.3.question=<div title="UW Statistics Bank/Continuous Distributions/Uniform Distributions/Q$Q">If X is a random variable with the (continuous) distribution shown below, find (3 decimal accuracy) P($BreakPt1 < X < $BreakPt2):
<p align="center"><applet width="443" height="165" code="applets.labelImage.LabelImage">
<param value="__BASE_URI__CPD/UD/Uniform.gif" name="image" />
<param value="1" name="size" />
<param value="33" name="label.1.x" />
<param value="67" name="label.1.y" />
<param value="$p" name="label.1.text" /> </applet></p>
</div>@
qu.7.3.answer.num=$Ans@
qu.7.3.answer.units=@
qu.7.3.showUnits=false@
qu.7.3.grading=toler_abs@
qu.7.3.err=0.001@
qu.7.3.negStyle=minus@
qu.7.3.numStyle=thousands scientific dollars arithmetic@
qu.7.3.mode=Numeric@
qu.7.3.name=03. Uniform, find P(n1 < X < n2)@
qu.7.3.comment=<p>Notice that the pdf is simply:<br />
<br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lcub;</mo><mfrac linethickness='0'><mi mathvariant='normal'>$p</mi><mn>0</mn></mfrac><mrow><mfrac linethickness='0'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>&alpha;</mi></mrow><mi mathvariant='normal'>otherwise</mi></mfrac></mrow></mrow></mrow></mstyle></math><br />
<br />
So to find &alpha; you must integrate this over [0,&alpha;], set that integral equal to 1 and solve to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>&#945;</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='thickmathspace' rspace='thickmathspace' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&equals;</mo><mfrac linethickness='1' denomalign='center' numalign='center' bevelled='false'><mn mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>1</mn><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$p</mi></mfrac></mrow><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='italic' fontfamily='Times New Roman'>or</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&InvisibleTimes;</mo><mi mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman'>$alpha</mi><mo mathcolor='#000000' mathbackground='#ffffff' mathsize='12' mathvariant='normal' fontfamily='Times New Roman' form='infix' fence='false' separator='false' lspace='0em' rspace='0em' stretchy='false' symmetric='false' maxsize='infinity' minsize='1' largeop='false' movablelimits='false' accent='false'>&period;</mo></mrow></math><br />
Now <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$BreakPt1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$BreakPt2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'></mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$BreakPt1</mi></mrow><mrow><mi mathvariant='normal'>$BreakPt2</mi></mrow></munderover><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi mathcolor='#800080'>x</mi><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'></mo><mi mathvariant='normal'>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$BreakPt2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$BreakPt1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'></mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'></mo><mi mathvariant='normal'>$Ans</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'></mo></mrow></mstyle></math></p>@
qu.7.3.editing=useHTML@
qu.7.3.solution=@
qu.7.3.algorithm=$Q=3;
$p=decimal(1,range(0.1,1/3,0.05));
$alpha=decimal(3,1/$p);
$BreakPt1 = range(1,int($alpha)-1,1);
$BreakPt2 = range($BreakPt1 + 1,int($alpha),1);
$Ans=decimal(4,$p*($BreakPt2 - $BreakPt1));@
qu.7.3.uid=a77d0867-3fd4-4d73-9d12-4810e384dc3f@
qu.7.3.info=  Course=230;
  Type=numeric;
@

