qu.1.topic=Population Standard Deviation Known@

qu.1.1.mode=Inline@
qu.1.1.name=Determine Confidence Level@
qu.1.1.comment=<p>The confidence level is given as the&nbsp;area under the standard normal distribution in between the values of -$zAlpha2 and $zAlpha2, which is graphically represented as:</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Using a standard normal table, we can find this area to be $ConfLevel.&nbsp; Therefore, the confidence level is $ConfLevel X 100% = $ConfPercent.</p>
<p>&nbsp;</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$zAlpha2=rand(1.01, 1.50, 3);
$Cumulative=maple("
X:=Statistics[CDF](Normal(0,1), $zAlpha2):
X
");
$ConfLevel=($Cumulative - 0.5)*2;
$ConfPercent=$ConfLevel*100;
$a=range(1.39,1.55,0.02);
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..-1*$zAlpha2, colour=blue): 
p2 := plot(f, x=-1*$zAlpha2..$zAlpha2, colour=blue, filled=true):
p3 := plot(f, x=$zAlpha2..3, colour=blue): 
p4 := plots[textplot]([-1*$zAlpha2, -0.05, `-$zAlpha2`], color=blue):
p5 := plots[textplot]([$zAlpha2, -0.05, `$zAlpha2`], color=blue):
plots[display]({p1,p2,p3,p4,p5}), plotoptions='width=350,height=350'
");@
qu.1.1.uid=b84272f7-69c7-447a-937b-40290d09d2aa@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.1.weighting=1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.answer.units=@
qu.1.1.part.1.numStyle=   @
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.err=0.01@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=toler_abs@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$ConfPercent@
qu.1.1.question=<p>For the confidence interval given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plusmn;</mo><mi>$zAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfrac><mrow><mi>&sigma;</mi></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, what is the corresponding confidence level?&nbsp; For the purposes of this question, we can assume that <em>n</em> is sufficiently large, 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>&sigma;</mi></mrow></mstyle></math>&nbsp;is known.</p><p>&nbsp;</p><p>Express your answer as a percent, but do NOT include the percent sign (%) in your response.</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 2 decimal places.</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Mean Rainfall: State hypotheses, test statistic, conclusion@
qu.1.2.comment=<p>a)&nbsp; Hypothesis testing is always carried out on the population parameter, which is this case is the population mean <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>.&nbsp; The null hypothesis is what the current belief is; here, it is that the mean rainfall is $mu mm.&nbsp; The alternative hypothesis is the new idea that the researcher believes, which in this case is that the mean rainfall is actually greater than $mu mm.&nbsp; Therefore, the appropriate null and alternative hypotheses are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</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><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&gt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To calculate the test statistic, we use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mrow><mfrac><mi>&sigma;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>.&nbsp;&nbsp; Substituting in the corresponding values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar</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>$mu</mi></mrow></mfenced><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ZTest</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; To determine what conclusion can be made, we need to calculate the p-value for the test statistic calculated in part (b).&nbsp; Because the alternative hypothesis indicates that we are carrying out a one-sided, upper tailed test, the p-value is the area under the standard normal curve to the right of our calculated test statistic.&nbsp; Using computer software, or a standard normal table, we can find this area to be $pvalue.&nbsp; Comparing the p-value to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, we see that our p-value is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi></mrow></mstyle></math>, indicating that there is <strong>insufficient evidence</strong> to reject the null hypothesis.</p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$mu=range(88, 92);
$sigma=rand(15, 20, 3);
$xbar=$mu + switch(rint(2), 2, 3);
$ZTest=($xbar - $mu)/($sigma/sqrt(10));
$fill=$ZTest*1.0;
$pvalue=1-erf($fill);
condition:gt($pvalue,0.10);@
qu.1.2.uid=b6c7ca6c-ec6c-4daa-96b1-ed8a3d664247@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.1.2.weighting=1,1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.fixed=@
qu.1.2.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</mi></mrow></mstyle></math>@
qu.1.2.part.1.question=null@
qu.1.2.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><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>$mu</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>@
qu.1.2.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mu</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><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&ne;</mo><mi>$mu</mi></mrow></mstyle></math>@
qu.1.2.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</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><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&gt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>@
qu.1.2.part.1.mode=Multiple Choice@
qu.1.2.part.1.display=vertical@
qu.1.2.part.1.answer=1@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.answer.units=@
qu.1.2.part.2.numStyle=   @
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.showUnits=false@
qu.1.2.part.2.err=0.0010@
qu.1.2.part.2.question=(Unset)@
qu.1.2.part.2.mode=Numeric@
qu.1.2.part.2.grading=toler_abs@
qu.1.2.part.2.negStyle=both@
qu.1.2.part.2.answer.num=$ZTest@
qu.1.2.part.3.name=sro_id_3@
qu.1.2.part.3.editing=useHTML@
qu.1.2.part.3.fixed=@
qu.1.2.part.3.question=null@
qu.1.2.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore no significant&nbsp;evidence that the mean rainfall in June is different from&nbsp;$mu mm.@
qu.1.2.part.3.choice.1=There is sufficient evidence to reject the null hypothesis, and therefore&nbsp;conclude that the mean rainfall in June is more than $mu mm.@
qu.1.2.part.3.mode=Multiple Choice@
qu.1.2.part.3.display=vertical@
qu.1.2.part.3.answer=2@
qu.1.2.question=<p>A local&nbsp;newscaster&nbsp;reports that the average rainfall in the month of June is approximately $mu mm.&nbsp; However, a meteorologist wishes to test this claim, believing that the average rainfall in June is actually higher than $mu mm.&nbsp; He collects data on the average June rainfall for 10 randomly selected years, and computes a mean of $xbar mm.&nbsp; Assuming that the population standard deviation is known to be $sigma, and that rainfall is normally distributed, determine each of the following:</p><p>&nbsp;</p><p>a)&nbsp; What are the appropriate hypotheses:</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Calculate the appropriate test statistic.</span></p><p>&nbsp;</p><p><span>Round your answer to&nbsp;at least 3 decimal places.</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; What is the appropriate conclusion that can be made, at the 5% level of significance?</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.1.3.mode=Inline@
qu.1.3.name=Calculate Power, sample mean, make comparison@
qu.1.3.comment=<p>a)&nbsp; To determine the value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover></mrow></mstyle></math>&nbsp;that would result in a <em>z</em> test statistic of $Zcutoff, we can rearrange the equation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&mu;</mi></mrow></mfenced><mrow><mfrac><mrow><mi>&sigma;</mi></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>&nbsp;to 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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi>&sigma;</mi></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&mu;</mi></mrow></mrow></mstyle></math>.&nbsp; Substituting in values for <em>Z</em>, <em>n</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>&sigma;</mi></mrow></mstyle></math>&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>&mu;</mi></mrow></mstyle></math>&nbsp;results in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Zcutoff</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$mu0</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; In order to calculate the Power of the test, we need to calculate the probability of rejecting the null hypothesis when it is false.&nbsp; Since the researcher stated she would reject the null hypothesis if she calculated a <em>z</em> test statistic greater than $Zcutoff, and in part (a) we calculated that this corresponded to a sample mean of $xbar, it is equivalent to say that the researcher will reject the null hypothesis if she obtains a sample mean greater than $xbar.</p>
<p>Therefore,&nbsp;to calculate the probability of rejecting the null hypothesis when it is false, we need to calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$xbar</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>when</mi><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>$muTrue</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$muTrue</mi></mrow></mfenced><mrow><mfrac><mrow><mi>$sigma</mi></mrow><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$ZStatistic</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Using a standard normal table, we can calculate this probability to be $Power, which is the Power of our test.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The Power of a test increases as the difference between the hypothesized mean and true mean increases.&nbsp; Therefore, if the difference between the hypothesized mean and true mean decreases, the Power will also decrease, as it becomes more difficult for the test to distinguish between the two values.</p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$mu0=range(101, 110);
$sigma=range(21, 26);
$n=range(45, 50);
$Zcutoff=range(1.1, 1.6, 2);
$xbar=($Zcutoff*($sigma/sqrt($n)))+$mu0;
$muTrue=$mu0+10;
$ZStatistic=($xbar-$muTrue)/($sigma/sqrt($n));
$Tail=maple("
X:=Statistics[CDF](Normal(0,1), $ZStatistic):
X
");
$Power=1-$Tail;
$muHalf=$mu0 + 5;@
qu.1.3.uid=9a3e82bc-9d3f-40ca-8503-8f5219f12a0b@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Application;
@
qu.1.3.weighting=1,1,1@
qu.1.3.numbering=alpha@
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qu.1.3.part.1.err=0.0010@
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qu.1.3.part.1.grading=toler_abs@
qu.1.3.part.1.negStyle=both@
qu.1.3.part.1.answer.num=$xbar@
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qu.1.3.part.2.grading=toler_abs@
qu.1.3.part.2.negStyle=both@
qu.1.3.part.2.answer.num=$Power@
qu.1.3.part.3.grader=exact@
qu.1.3.part.3.name=sro_id_3@
qu.1.3.part.3.editing=useHTML@
qu.1.3.part.3.display.permute=true@
qu.1.3.part.3.question=(Unset)@
qu.1.3.part.3.answer.2=Lower@
qu.1.3.part.3.answer.1=Higher@
qu.1.3.part.3.mode=List@
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qu.1.3.part.3.credit.2=1.0@
qu.1.3.part.3.credit.1=0.0@
qu.1.3.question=<p>A researcher wishes to test the null hypothesis that the mean of a normally distributed population is equal to $mu0, against the alternative hypothesis that the mean is&nbsp;greater than&nbsp;$mu0.&nbsp;&nbsp;She is not that familiar with hypothesis testing, so she decides to randomly select a sample of size $n, and if the <em>Z</em> test statistic is greater than $Zcutoff, she will reject the null hypothesis.&nbsp; Assuming that the population standard deviation is known to be $sigma, then:</p><p>&nbsp;</p><p>a)&nbsp; What value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover></mrow></mstyle></math>&nbsp;would she need to obtain in order to get a <em>Z</em> test statistic of $Zcutoff?</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Suppose that the <em>true</em> population mean was actually $muTrue.&nbsp; What is the value of the power for the test?</span></p><p>&nbsp;</p><p><span>Round your responses to at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; What if, instead of $muTrue, the true population mean was only $muHalf.&nbsp; Would you expect the power of the test to be higher or lower than the value you found in part (b)?</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=Calculate one-sided p-value (2)@
qu.1.4.comment=<p>a)&nbsp; Since the alternative hypothesis indicates a one-sided, upper tailed test, the p-value is the area under the standard normal curve to the right of the test statistic.&nbsp; Using computer software, or approximating with a standard normal table, we can find this area to be <em>p-value = $pvalue</em>.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>b)&nbsp; Since&nbsp;the p-value is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis, at the 10% level of significance.</p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$z=rand(1.3, 1.6, 3);
$Tail=erf($z);
$pvalue=1-$Tail;
$pvaluedisplay=decimal(4, $pvalue);
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$z, colour=blue): 
p2 := plot(f, x=$z..3, colour=blue, filled=true): 
p3 := plots[textplot]([$z, -0.05, `$z`], color=blue):
p4 := plots[textplot]([2.0, 0.04, `$pvaluedisplay`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");
condition:lt($pvalue,0.10);@
qu.1.4.uid=a7689fcc-0c9f-40dd-b038-b2bded6f6ed9@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
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qu.1.4.part.1.negStyle=both@
qu.1.4.part.1.answer.num=$pvalue@
qu.1.4.part.2.grader=exact@
qu.1.4.part.2.name=sro_id_2@
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qu.1.4.part.2.display.permute=true@
qu.1.4.part.2.question=(Unset)@
qu.1.4.part.2.answer.2=No@
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qu.1.4.question=<p>Suppose a hypothesis test of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>is tested against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&gt;</mo><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></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>, and the resulting <em>Z</em> test statistic 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>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><mi>$z</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the appropriate p-value for the hypothesis test.</p><p>&nbsp;</p><p>Round your response to at least&nbsp;4 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Based on the p-value, would you reject the null hypothesis at the 10% level of significance?</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.5.mode=Inline@
qu.1.5.name=Calculate P(Type I Error), corresponding sample means.@
qu.1.5.comment=<p>a)&nbsp; Type I error occurs when the null hypothesis is rejected when it is actually true.&nbsp; Since the researcher will reject the null hypothesis if she calculates a <em>z</em> test statistic greater than $Zcutoff or less than -$Zcutoff, the probability of a Type I error is calculated 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>P</mi><mfenced open='(' close=')' separators=','><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$Zcutoff</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Using a standard normal table, we can find this area to be $Error.&nbsp; Therefore, <em>P(Type I error) = $Error.</em></p>
<p>&nbsp;</p>
<p>b)&nbsp; To find the values of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover></mrow></mstyle></math>&nbsp;that correspond to a <em>z</em> test statistic value of +/- $Zcutoff, we can rearrange the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&mu;</mi></mrow></mfenced><mrow><mfrac><mi>&sigma;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>&nbsp;to 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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>&sigma;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&mu;</mi></mrow></mrow></mstyle></math>.&nbsp; Because there are two possible <em>z</em> test statistic values, we need to calculate two <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover></mrow></mstyle></math>&nbsp;values: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Zcutoff</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$mu</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi></mrow></mstyle></math>, 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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Zcutoff</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$mu</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math>.&nbsp; Therefore, corresponding sample means are $xbar2 and $xbar1.</p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$mu=range(101, 110);
$sigma=range(21, 26);
$n=range(45, 50);
$Zcutoff=range(1.1, 1.6, 2);
$Tail=maple("
X:=Statistics[CDF](Normal(0,1), -1*$Zcutoff):
X
");
$Error = 2*$Tail;
$xbar1=($Zcutoff*($sigma/sqrt($n)))+$mu;
$xbar2=(-1*$Zcutoff*($sigma/sqrt($n)))+$mu;@
qu.1.5.uid=fa3e26b3-06c5-4d83-8488-16a642813fdf@
qu.1.5.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Application;
@
qu.1.5.weighting=1,1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.answer.units=@
qu.1.5.part.1.numStyle=   @
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.showUnits=false@
qu.1.5.part.1.err=0.0010@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.mode=Numeric@
qu.1.5.part.1.grading=toler_abs@
qu.1.5.part.1.negStyle=both@
qu.1.5.part.1.answer.num=$Error@
qu.1.5.part.2.editing=useHTML@
qu.1.5.part.2.question=(Unset)@
qu.1.5.part.2.name=sro_id_2@
qu.1.5.part.2.answer=$xbar2?0.01, $xbar1?0.01@
qu.1.5.part.2.mode=Ntuple@
qu.1.5.question=<p>A researcher wishes to test the null hypothesis that the mean of a normally distributed population is equal to $mu, against the alternative hypothesis that the mean is not equal to $mu.&nbsp; He is not that familiar with hypothesis testing, so he decides to randomly select a sample of size $n, and if the <em>Z</em> test statistic is greater than $Zcutoff or less than -$Zcutoff, he will reject the null hypothesis.&nbsp; Assuming that the population standard deviation is known to be $sigma, then:</p><p>&nbsp;</p><p>a)&nbsp; What is the probability he will make a Type I error?</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What are the corresponding <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover></mrow></mstyle></math>&nbsp;values associated with his arbitrarily selected rejection region?&nbsp; That is, what values 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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover></mrow></mstyle></math>&nbsp;will result in a <em>Z</em> test statistic of +/- $Zcutoff?</span></p><p>&nbsp;</p><p><span>Round your responses to at least&nbsp;3 decimal places.</span></p><p><span>Enter your responses in the format: <strong><em>Smaller Value, Larger Value&nbsp; </em></strong>(include the ',' between the values).</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.6.mode=Inline@
qu.1.6.name=Determine Confidence Coefficient@
qu.1.6.comment=<p>In order to determine the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>values, we need to capture the middle $CLPercent % of the standard normal distribution.&nbsp; That is, we need to find a <em>z</em> value such that the area between <em>-z</em> and <em>z</em> is $CLDecimal:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Using a standard normal table, we can find the <em>z</em> value to be $zAlpha2.</p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$CLPercent=rand(70, 80, 3);
$CLDecimal=$CLPercent/100;
$QTL=($CLDecimal/2) + 0.5;
$zAlpha2=maple("
X:=Statistics[Quantile](Normal(0,1), $QTL):
X
");
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..-1*$zAlpha2, colour=blue): 
p2 := plot(f, x=-1*$zAlpha2..$zAlpha2, colour=blue, filled=true):
p3 := plot(f, x=$zAlpha2..3, colour=blue):
p4 := plots[textplot]([0.1, 0.15, `<-$CLDecimal->`], color=black):
p5 := plots[textplot]([-1*$zAlpha2, -0.05, `-z`], color=blue):
p6 := plots[textplot]([$zAlpha2, -0.05, `z`], color=blue):
plots[display]({p1,p2,p3,p4,p5,p6}), plotoptions='width=350,height=350'
");@
qu.1.6.uid=dc3ea52b-20eb-4329-933c-9b93ee860191@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.6.weighting=1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.answer.units=@
qu.1.6.part.1.numStyle=   @
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.showUnits=false@
qu.1.6.part.1.err=0.01@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.mode=Numeric@
qu.1.6.part.1.grading=toler_abs@
qu.1.6.part.1.negStyle=both@
qu.1.6.part.1.answer.num=$zAlpha2@
qu.1.6.question=<p>What is the appropriate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mrow><mi>&alpha;</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>value for a $CLPercent % confidence interval 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>&mu;</mi></mrow></mstyle></math>?</p><p>&nbsp;</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 2 decimal places.</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Calculate two-sided test statistic, p-value, conclusion@
qu.1.7.comment=<p>a)&nbsp; The <em>z</em> test statistic is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mfrac><mi>&sigma;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate the test statistic to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>$xbar</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mu</mi></mrow><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ZTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are performing a two-sided test.&nbsp; Therefore, the p-value is found as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$ZTest</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>.&nbsp; Using computer software, or approximating with a standard normal table, we can 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><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$ZTest</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$UpperTail</mi></mrow></mstyle></math>, and therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>value</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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$UpperTail</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$pvalue</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c) i)&nbsp; Since the p-value is larger than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis at the 5% level of significance.</p>
<p>ii) Since the p-value is larger than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis at the 5% level of significance.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$n=range(35, 40);
$mu=rand(3, 4, 2);
$xbar = $mu + 0.2;
$ZTest=($xbar - $mu)/(0.89/sqrt($n));
$LowTail=erf($ZTest);
$UpperTail=1-$LowTail;
$pvalue=2*$UpperTail;
condition:gt($pvalue,0.10);@
qu.1.7.uid=6230844a-ab51-46df-8119-3a13c16c3982@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.7.weighting=1,1,1,1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.answer.units=@
qu.1.7.part.1.numStyle=   @
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.showUnits=false@
qu.1.7.part.1.err=0.0010@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.mode=Numeric@
qu.1.7.part.1.grading=toler_abs@
qu.1.7.part.1.negStyle=both@
qu.1.7.part.1.answer.num=$ZTest@
qu.1.7.part.2.name=sro_id_2@
qu.1.7.part.2.answer.units=@
qu.1.7.part.2.numStyle=   @
qu.1.7.part.2.editing=useHTML@
qu.1.7.part.2.showUnits=false@
qu.1.7.part.2.err=0.01@
qu.1.7.part.2.question=(Unset)@
qu.1.7.part.2.mode=Numeric@
qu.1.7.part.2.grading=toler_abs@
qu.1.7.part.2.negStyle=both@
qu.1.7.part.2.answer.num=$pvalue@
qu.1.7.part.3.grader=exact@
qu.1.7.part.3.name=sro_id_3@
qu.1.7.part.3.editing=useHTML@
qu.1.7.part.3.display.permute=true@
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qu.1.7.question=<p>Test the null hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mu</mi></mrow></mstyle></math>against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&ne;</mo><mi>$mu</mi></mrow></mstyle></math>, based on a random sample of $n observations drawn from a normally distributed population with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math>&nbsp;and &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>&sigma;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.89</mn></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the test statistic?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the appropriate p-value?</span></p><p>&nbsp;</p><p><span>Round your response to at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Is the null hypothesis rejected at:</span></span></p><p><span><span>i)&nbsp; the 5% level of significance?&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p><span><span><span>ii)&nbsp; the 10% level of significance?&nbsp;&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span></span></span></span></p>@

qu.1.8.mode=Inline@
qu.1.8.name=Determining one-sided rejection region@
qu.1.8.comment=<p>Since the alternative hypothesis is a one-sided, lower tailed hypothesis, the critical value of <em>z</em> is one such that the area under the standard normal curve to the left of <em>z</em> must be $alphaRound.&nbsp; Using a standard normal table, we can find this critical value to be $zCritical.</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$zCritical=rand(-1.3, -1.01, 3);
$alphaRaw=maple("
X:=Statistics[CDF](Normal(0,1), $zCritical):
X
");
$alphaRound=numfmt("0.0000", $alphaRaw);
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$zCritical, colour=blue, filled=true): 
p2 := plot(f, x=$zCritical..3, colour=blue): 
p3 := plots[textplot]([$zCritical, -0.05, `$zCritical`], color=blue):
p4 := plots[textplot]([-2.0, 0.02, `$alphaRound`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");@
qu.1.8.uid=9786b551-4bd0-4eda-b61f-0133d1fa9f01@
qu.1.8.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.8.weighting=1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.name=sro_id_1@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.fixed=@
qu.1.8.part.1.question=null@
qu.1.8.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>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$zCritical</mi></mrow></mstyle></math>@
qu.1.8.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$zCritical</mi></mrow></mstyle></math>@
qu.1.8.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><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mi>Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$zCritical</mi></mrow></mstyle></math>@
qu.1.8.part.1.mode=Multiple Choice@
qu.1.8.part.1.display=vertical@
qu.1.8.part.1.answer=3@
qu.1.8.question=<p>Suppose a hypothesis test&nbsp;of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>is being carried out against the one-sided alternative&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><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&lt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math> at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$alphaRound</mi></mrow></mstyle></math>.&nbsp; Assume that the population is normally distributed, and 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>&sigma;</mi></mrow></mstyle></math>&nbsp;is known.</p><p>&nbsp;</p><p>What is the appropriate rejection region?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.9.mode=Inline@
qu.1.9.name=Calculating Margin of Error for 90% & 95% CL@
qu.1.9.comment=<p>a)&nbsp; The margin of error is&nbsp;calculated by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>z</mi><mrow><mfrac><mrow><mi>&alpha;</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfrac><mrow><mi>&sigma;</mi></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; For a 95% confidence level, the corresponding <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>value is 1.96.&nbsp; Therefore, the margin of error 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><mn>1.96</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME1</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Using the same formula for margin of error as above, but with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>= 1.645 for a 90% confidence level,&nbsp;the calculated margin of error is then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1.645</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME2</mi></mrow></mstyle></math>.&nbsp;</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$n=range(30, 40);
$sigma=rand(5, 9, 2);
$ME1=1.96*($sigma/sqrt($n));
$ME2=1.645*($sigma/sqrt($n));@
qu.1.9.uid=d5857124-b87b-449d-9cc5-eacf321b2694@
qu.1.9.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.9.weighting=1,1@
qu.1.9.numbering=alpha@
qu.1.9.part.1.name=sro_id_1@
qu.1.9.part.1.answer.units=@
qu.1.9.part.1.numStyle=   @
qu.1.9.part.1.editing=useHTML@
qu.1.9.part.1.showUnits=false@
qu.1.9.part.1.err=0.0010@
qu.1.9.part.1.question=(Unset)@
qu.1.9.part.1.mode=Numeric@
qu.1.9.part.1.grading=toler_abs@
qu.1.9.part.1.negStyle=both@
qu.1.9.part.1.answer.num=$ME1@
qu.1.9.part.2.name=sro_id_2@
qu.1.9.part.2.answer.units=@
qu.1.9.part.2.numStyle=   @
qu.1.9.part.2.editing=useHTML@
qu.1.9.part.2.showUnits=false@
qu.1.9.part.2.err=0.0010@
qu.1.9.part.2.question=(Unset)@
qu.1.9.part.2.mode=Numeric@
qu.1.9.part.2.grading=toler_abs@
qu.1.9.part.2.negStyle=both@
qu.1.9.part.2.answer.num=$ME2@
qu.1.9.question=<p>Suppose a random sample of size $n was drawn from a normally distributed population, with a known population standard deviation of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$sigma</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the margin of error for a 95% confidence level?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3 decimal places.</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span>b)&nbsp; What is the margin of error for a 90% confidence level?</span></p><p>&nbsp;</p><p><span>Round your response to&nbsp;at least 3 decimal places.</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p>@

qu.1.10.mode=Inline@
qu.1.10.name=Determine Confidence Coefficient and Margin of Error@
qu.1.10.comment=<p>a)&nbsp; The&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><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>value for a $CLPercent % confidence interval is $Quant.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Using the formula for margin of error, with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>=&nbsp;$Quant&nbsp;for a $CLPercent % confidence level,&nbsp;we get&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$Quant</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp;</p>@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$n=range(80, 90);
$sigma=rand(10, 15, 3);
$CLPercent=range(50, 60);
$CLDecimal=$CLPercent/100;
$QTL=($CLDecimal/2) + 0.5;
$Quant=maple("
X:=Statistics[Quantile](Normal(0,1), $QTL):
X
");
$ME=$Quant*($sigma/sqrt($n));@
qu.1.10.uid=f8646af7-8fb7-40d4-a82e-b26af2460598@
qu.1.10.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.10.weighting=1,1@
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qu.1.10.part.1.numStyle=   @
qu.1.10.part.1.editing=useHTML@
qu.1.10.part.1.showUnits=false@
qu.1.10.part.1.err=0.01@
qu.1.10.part.1.question=(Unset)@
qu.1.10.part.1.mode=Numeric@
qu.1.10.part.1.grading=toler_abs@
qu.1.10.part.1.negStyle=both@
qu.1.10.part.1.answer.num=$Quant@
qu.1.10.part.2.name=sro_id_2@
qu.1.10.part.2.answer.units=@
qu.1.10.part.2.numStyle=   @
qu.1.10.part.2.editing=useHTML@
qu.1.10.part.2.showUnits=false@
qu.1.10.part.2.err=0.01@
qu.1.10.part.2.question=(Unset)@
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qu.1.10.question=<p>Suppose a random sample of size $n was drawn from a normally distributed population, with a known population standard deviation of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$sigma</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the&nbsp;appropriate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>&nbsp;value&nbsp;for a $CLPercent % confidence interval?</p><p>&nbsp;</p><p>Round your response to at least&nbsp;2 decimal places.</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the margin of error for a $CLPercent % confidence interval?</span></p><p>&nbsp;</p><p><span>Round your response to&nbsp;at least 2 decimal places.</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p>@

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qu.1.11.name=Calculate one-sided test statistic, p-value, conclusion (2)@
qu.1.11.comment=<p>a)&nbsp; The test statistic can be calculated using the formula: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><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>&mu;</mi></mrow></mfenced><mrow><mfrac><mi>&sigma;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</mi></mrow></mfenced><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ZTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Since the alternative hypothesis indicates that the test is a one-sided, lower tailed test, the p-value is calculated as the area under the standard normal curve to the <em>left</em> of the test statistic.&nbsp; Therefore,&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>p</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>value</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$ZTest</mi></mrow></mfenced></mrow><mrow><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>$pvalue</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c) i)&nbsp; Since the p-value calculated in part (b) is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, the null hypothesis <em>is</em> rejected at the 10% level of significance.</p>
<p>ii)&nbsp; At the 5% level of significance, the p-value is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, and therefore the null hypothesis is <em>not</em> rejected.</p>@
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qu.1.11.algorithm=$sigma=rand(0.65, 0.75, 2);
$mu=rand(3, 4, 2);
$xbar = $mu - 0.2;
$ZTest=($xbar - $mu)/($sigma/sqrt(25));
$pvalue=maple("
X1:=Statistics[CDF](Normal(0,1), $ZTest):
X1
");
condition:gt($pvalue,0.05);
condition:lt($pvalue,0.10);@
qu.1.11.uid=c08df830-c049-4d77-bc67-a9f6de657791@
qu.1.11.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.11.question=<p>Test the null hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mu</mi></mrow></mstyle></math>against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&lt;</mo><mi>$mu</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, based on a random sample of&nbsp;25 observations drawn from a normally distributed population with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math>&nbsp;and&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>&sigma;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sigma</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the test statistic?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the appropriate p-value?</span></p><p>&nbsp;</p><p><span>Round your response to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Is the null hypothesis rejected at:</span></span></p><p><span><span>i)&nbsp; the 10% level of significance?&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p><span><span><span>ii)&nbsp; the 5% level of significance?&nbsp;&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span></span></span></span></p>@

qu.1.12.mode=Inline@
qu.1.12.name=Calculate one-sided p-value (1)@
qu.1.12.comment=<p>a)&nbsp; The alternative hypothesis indicates that we are performing a one-sided, lower tailed test.&nbsp; Therefore, the p-value is the area under the standard normal curve to the left of the test statistic.&nbsp; Using computer software, or approximating with a standard normal table, we can find this area to be <em>p-value = $pvalue</em>.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>b)&nbsp; Since the p-value is great than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis, at the 10% level of significance.</p>@
qu.1.12.editing=useHTML@
qu.1.12.solution=@
qu.1.12.algorithm=$z=rand(1.01, 1.6, 3);
$pvalue=erf($z);
$pvaluedisplay=decimal(4, $pvalue);
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$z, colour=blue, filled=true): 
p2 := plot(f, x=$z..3, colour=blue): 
p3 := plots[textplot]([$z, -0.05, `$z`], color=blue):
p4 := plots[textplot]([-1.0, 0.04, `$pvaluedisplay`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");@
qu.1.12.uid=b3b74ecf-9170-4f4b-bdf8-74c730dcfa86@
qu.1.12.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
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qu.1.12.question=<p>Suppose a hypothesis test of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>is tested against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&lt;</mo><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, and the resulting <em>Z</em> test statistic 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>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><mi>$z</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the appropriate p-value for the hypothesis test.</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 4 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Based on the p-value, would you reject the null hypothesis at the 10% level of significance?</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.13.mode=Inline@
qu.1.13.name=Bass Fishing: Calculating point estimate, confidence interval for mean@
qu.1.13.comment=<p>a)&nbsp; A point estimate for the population mean, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>, is the sample estimate, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover></mrow></mstyle></math>.&nbsp; Therefore, the point estimate 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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>10</mn></mrow><mrow><mn>10</mn></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$Obs1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$Obs2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$Obs10</mi></mrow></mfenced><mrow><mn>10</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The formula for 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><mfenced open='(' close=')' separators=','><mrow><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>&alpha;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>100</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>%</mi></mrow></mstyle></math>&nbsp;confidence interval 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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plusmn;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfrac><mi>&sigma;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; For a 90% confidence interval, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>= 1.645.&nbsp; Plugging all the values into the equation gives us <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>$xbar</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plusmn;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1.645</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>&nbsp;.&nbsp; Written in interval notation, the 90% confidence interval 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>&mu;</mi></mrow></mstyle></math>&nbsp;is ($LowerCL, $UpperCL).</p>@
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$Obs1=rand(12, 18, 3);
$Obs2=rand(12, 18, 3);
$Obs3=rand(12, 18, 3);
$Obs4=rand(12, 18, 3);
$Obs5=rand(12, 18, 3);
$Obs6=rand(12, 18, 3);
$Obs7=rand(12, 18, 3);
$Obs8=rand(12, 18, 3);
$Obs9=rand(12, 18, 3);
$Obs10=rand(12, 18, 3);
$sigma=rand(1.01, 1.5, 3);
$xbar=($Obs1+$Obs2+$Obs3+$Obs4+$Obs5+$Obs6+$Obs7+$Obs8+$Obs9+$Obs10)/10;
$SE=$sigma/sqrt(10);
$ME=1.645*$SE;
$LowerCL=$xbar - $ME;
$UpperCL=$xbar + $ME;@
qu.1.13.uid=077fb4a4-7e0c-46f9-a565-b48fb5fda7ab@
qu.1.13.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.1.13.weighting=1,1@
qu.1.13.numbering=alpha@
qu.1.13.part.1.name=sro_id_1@
qu.1.13.part.1.answer.units=@
qu.1.13.part.1.numStyle=   @
qu.1.13.part.1.editing=useHTML@
qu.1.13.part.1.showUnits=false@
qu.1.13.part.1.question=(Unset)@
qu.1.13.part.1.mode=Numeric@
qu.1.13.part.1.grading=exact_value@
qu.1.13.part.1.negStyle=both@
qu.1.13.part.1.answer.num=$xbar@
qu.1.13.part.2.editing=useHTML@
qu.1.13.part.2.question=(Unset)@
qu.1.13.part.2.name=sro_id_2@
qu.1.13.part.2.answer=($LowerCL?0.01, $UpperCL?0.01)@
qu.1.13.part.2.mode=Ntuple@
qu.1.13.question=<p>Suppose in a popular fishing lake, 10 largemouth bass are caught, and each of their lengths (in inches)&nbsp;are recorded below:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>Fish</td>            <td>#1</td>            <td>#2</td>            <td>#3</td>            <td>#4</td>            <td>#5</td>            <td>#6</td>            <td>#7</td>            <td>#8</td>            <td>#9</td>            <td>#10</td>        </tr>        <tr>            <td>Length (inches)</td>            <td>$Obs1</td>            <td>$Obs2</td>            <td>$Obs3</td>            <td>$Obs4</td>            <td>$Obs5</td>            <td>$Obs6</td>            <td>$Obs7</td>            <td>$Obs8</td>            <td>$Obs9</td>            <td>$Obs10</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>Furthermore, suppose it is known that the population standard deviation for length is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$sigma</mi></mrow></mstyle></math>, and that the length of largemouth bass is normally distributed.</p><p>&nbsp;</p><p>a)&nbsp; What is a point estimate of the population mean length?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 2 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is a 90% confidence interval for the population mean length?</span></p><p>&nbsp;</p><p><span>Enter your response in the interval notation: <strong>(lower limit, upper limit)</strong> .&nbsp; Include the brackets in your response.</span></p><p>&nbsp;</p><p><span>Round your values for lower limit and upper limit to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.14.mode=Inline@
qu.1.14.name=Determining two-sided rejection region@
qu.1.14.comment=<p>Since the alternative hypothesis is two-sided, there are two rejection regions (one in the lower tail, one in the upper tail), 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>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$alpha</mi></mrow></mstyle></math>divided equally between each region.&nbsp; The critical <em>z</em> value is then such that the area below <em>-z</em> and the area above <em>z</em> are each equal to $alpha2.&nbsp; Using a standard normal table, we can find the critical <em>z</em> value to be $true.</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.1.14.editing=useHTML@
qu.1.14.solution=@
qu.1.14.algorithm=$k=rint(2);
$alpha=switch($k, 0.10, 0.05);
$alpha2=$alpha/2;
$true=switch($k, 1.645, 1.96);
$distract=switch($k, 1.96, 1.645);
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..-1*$true, colour=blue, filled=true): 
p2 := plot(f, x=-1*$true..$true, colour=blue):
p3 := plot(f, x=$true..3, colour=blue, filled=true): 
p4 := plots[textplot]([-1*$true, -0.05, `-$true`], color=blue):
p5 := plots[textplot]([$true, -0.05, `$true`], color=blue):
p6 := plots[textplot]([-2.2, 0.02, `$alpha2`], color=black):
p7 := plots[textplot]([2.2, 0.02, `$alpha2`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6,p7}), plotoptions='width=350,height=350'
");@
qu.1.14.uid=2ce3a845-9bba-4224-b907-1e57d47bc7a2@
qu.1.14.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.14.weighting=1@
qu.1.14.numbering=alpha@
qu.1.14.part.1.name=sro_id_1@
qu.1.14.part.1.editing=useHTML@
qu.1.14.part.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>$true</mi></mrow></mstyle></math>@
qu.1.14.part.1.fixed=@
qu.1.14.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>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$distract</mi></mrow></mstyle></math>@
qu.1.14.part.1.question=null@
qu.1.14.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$true</mi></mrow></mstyle></math>@
qu.1.14.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><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mi>Z</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$distract</mi></mrow></mstyle></math>@
qu.1.14.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mi>Z</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$true</mi></mrow></mstyle></math>@
qu.1.14.part.1.mode=Multiple Choice@
qu.1.14.part.1.display=vertical@
qu.1.14.part.1.answer=1@
qu.1.14.question=<p>Suppose a hypothesis test&nbsp;of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>is being carried out against a two-sided alternative hypothesis at <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$alpha</mi></mrow></mstyle></math>.&nbsp; Assume that the population is normally distributed, and 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>&sigma;</mi></mrow></mstyle></math>&nbsp;is known.</p><p>&nbsp;</p><p>What is the appropriate rejection region?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.15.mode=Inline@
qu.1.15.name=Determining sample size for given CL, ME, and making comparisons@
qu.1.15.comment=<p>a)&nbsp; To estimate the sample size, we can re-arrange the formula for margin of error to 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><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msub><mi>z</mi><mrow><mfrac><mrow><mi>&alpha;</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&sigma;</mi></mrow></mrow><mrow><mi>ME</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>.&nbsp; Plugging in the given values, and the&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><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>value for a 95% confidence level, 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><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mn>1.96</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$sigma</mi></mrow><mrow><mi>$ME</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$nDecimal</mi></mrow></mstyle></math>.&nbsp; However, we cannot take a fraction of an individual, so our sample size must be rounded UP to the nearest whole number.&nbsp; Therefore, the minimum sample size required is <em>n</em>&nbsp; = $nFinal.</p>
<p>&nbsp;</p>
<p>b)&nbsp; If everything else were to remain constant, and only the population standard deviation were to increase, we can see from the formula above that this would result in an increase in the numerator, and consequently the minimum sample size would also increase.&nbsp; That is to say, if the variation in our population were to increase, then we would need a larger sample to be within the same margin of error of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>, with the same level of confidence.</p>
<p>&nbsp;</p>
<p>c)&nbsp; If everything else were to remain constant, and only the confidence level were to decrease, then the value 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><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>would decrease.&nbsp; This would result in a decrease in the numerator, and a subsequent decrease in the minimum sample size.&nbsp; That is to say, if we wanted to be within the same margin of error of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>, but with less confidence, then we could use a smaller sample.</p>
<p>&nbsp;</p>
<p>d)&nbsp; If the bound in which we wanted to estimate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&nbsp;(i.e. the margin of error) were to increase, then the denominator in our formula for sample size would increase, which results in an decrease in the minimum sample size required.&nbsp; That is to say, if we wanted to be within a larger margin of error of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>, but with the same confidence level, we could take a smaller sample.</p>@
qu.1.15.editing=useHTML@
qu.1.15.solution=@
qu.1.15.algorithm=$ME=range(2, 3);
$sigma=rand(6, 8, 2);
$nDecimal=(1.96*$sigma/$ME)^2;
$nFinal=maple("ceil($nDecimal)");@
qu.1.15.uid=ca1833a9-f7e9-4071-ad8f-31d83f0760d5@
qu.1.15.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Sample Size Determination;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.15.question=<p>a)&nbsp; In order to estimate the population mean, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>, to within $ME&nbsp;at 95% confidence, what is the minimum sample size required?&nbsp; (Assume<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$sigma</mi></mrow></mstyle></math>).</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; If just&nbsp;the population standard deviation were to increase, then the minimum sample size required would:&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p>c)&nbsp;&nbsp;If just&nbsp;the confidence level were to decrease (i.e. go from 95% to 90% confidence), then the minimum sample size required&nbsp;would:&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>d)&nbsp; If just the bound within which <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&nbsp;was to be&nbsp;estimated were to increase, then the minimum sample size required would:&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span>&nbsp;</span></p>@

qu.1.16.mode=Inline@
qu.1.16.name=Calculate two-sided p-value@
qu.1.16.comment=<p>a)&nbsp; Since we are performing a two-sided hypothesis test, the p-value is estimated 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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</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><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mfenced></mrow></mstyle></math>.&nbsp; Using computer software, or approximating from a standard normal table, we can find the p-value to be <em>2 X $UpperTail = $pvalue.</em>&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>b)&nbsp; Since the p-value is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&alpha;</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><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis, at the 5% level of significance.</p>@
qu.1.16.editing=useHTML@
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qu.1.16.algorithm=$z=rand(1.01, 1.5, 3);
$LowTail=erf($z);
$UpperTail=1-$LowTail;
$UpperDisplay=decimal(4, $UpperTail);
$pvalue=2*$UpperTail;
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..-1*$z, colour=blue, filled=true): 
p2 := plot(f, x=-1*$z..$z, colour=blue):
p3 := plot(f, x=$z..3, colour=blue, filled=true):  
p4 := plots[textplot]([-1*$z, -0.05, `-$z`], color=blue):
p5 := plots[textplot]([$z, -0.05, `$z`], color=blue):
p6 := plots[textplot]([2, 0.04, `$UpperDisplay`], color=black):
p7 := plots[textplot]([-2, 0.04, `$UpperDisplay`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6,p7}), plotoptions='width=350,height=350'
");@
qu.1.16.uid=e5f2dc9c-aba7-4b05-9a06-2cb0dcfe7a39@
qu.1.16.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
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qu.1.16.question=<p>Suppose a hypothesis test of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>is tested against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&ne;</mo></mrow><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>, and the resulting <em>Z</em> test statistic 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>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><mi>$z</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the appropriate p-value for the hypothesis test.</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 4 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Based on the p-value, would you reject the null hypothesis at the 5% level of significance?</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

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qu.1.17.name=Calculate one-sided test statistic, p-value, conclusion (1)@
qu.1.17.comment=<p>a)&nbsp; The formula for the <em>z</em> test statistic is given 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><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><mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><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><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mfrac><mi>&sigma;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate the <em>z</em> test statistic to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfrac><mrow><mi>$xbar</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</mi></mrow><mrow><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ZTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Since the alternative hypothesis indicates that we are perfoming a one-sided, upper tailed test, the p-value is determined as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$ZTest</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Using computer software, or approximating with a standard normal table, we can find this area to be <em>p-value = $pvalue</em>.</p>
<p>&nbsp;</p>
<p>c) i)&nbsp; Since the p-value is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis at the 10% level of significance.</p>
<p>ii)&nbsp; Since the p-value is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis at the 5% level of significance.</p>@
qu.1.17.editing=useHTML@
qu.1.17.solution=@
qu.1.17.algorithm=$n=range(65, 75);
$mu=rand(3, 4, 2);
$xbar = $mu + 0.2;
$ZTest=($xbar - $mu)/(0.89/sqrt($n));
$Tail=erf($ZTest);
$pvalue=1-$Tail;
condition:lt($pvalue,0.05);@
qu.1.17.uid=c55d35ad-92bb-4078-a472-38d5d26391fc@
qu.1.17.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Known, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.17.weighting=1,1,1,1@
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qu.1.17.part.1.editing=useHTML@
qu.1.17.part.1.showUnits=false@
qu.1.17.part.1.err=0.0010@
qu.1.17.part.1.question=(Unset)@
qu.1.17.part.1.mode=Numeric@
qu.1.17.part.1.grading=toler_abs@
qu.1.17.part.1.negStyle=both@
qu.1.17.part.1.answer.num=$ZTest@
qu.1.17.part.2.name=sro_id_2@
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qu.1.17.part.2.editing=useHTML@
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qu.1.17.part.2.grading=toler_abs@
qu.1.17.part.2.negStyle=both@
qu.1.17.part.2.answer.num=$pvalue@
qu.1.17.part.3.grader=exact@
qu.1.17.part.3.name=sro_id_3@
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qu.1.17.part.3.display.permute=true@
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qu.1.17.part.3.answer.2=No@
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qu.1.17.question=<p>Test the null hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mu</mi></mrow></mstyle></math>against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&gt;</mo><mi>$mu</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, based on a random sample of $n observations drawn from a normally distributed population with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math>&nbsp;and&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>&sigma;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.89</mn></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the test statistic?</p><p>&nbsp;</p><p>Round your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the appropriate p-value?</span></p><p>&nbsp;</p><p><span>Round your response to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Is the null hypothesis rejected at:</span></span></p><p><span><span>i)&nbsp; the 10% level of significance?&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p><span><span><span>ii)&nbsp; the 5% level of significance?&nbsp;&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span></span></span></span></p>@

qu.2.topic=Population Standard Deviation Unknown@

qu.2.1.mode=Inline@
qu.2.1.name=Determine one-sided rejection region (1)@
qu.2.1.comment=<p>A one-sided, upper tailed hypothesis test indicates that the rejection region is determined by a value of&nbsp;<em>T</em> such 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>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.01</mn></mrow></mstyle></math>, under a <em>t</em> distribution with $df degrees of freedom.&nbsp; Using computer software, or approximating with a <em>t</em> distribution table, we can find this value to be&nbsp;<em>T = $TrueRaw</em>.&nbsp; Therefore, the rejection region is approximately&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$True</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, which is represented graphically as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$n=range(5,10);
$df=$n-1;
$TrueRaw=invstudentst($df, 0.99);
$True=numfmt("#.000", $TrueRaw);
$DistractRaw=invstudentst($df, 0.995);
$Distract=numfmt("#.000", $DistractRaw);
$p=plotmaple("
f := Statistics[PDF](StudentT($df),x): 
p1 := plot(f, x=-3..$TrueRaw, colour=blue): 
p2 := plot(f, x=$TrueRaw..3, colour=blue, filled=true): 
p3 := plots[textplot]([$TrueRaw, -0.05, `$True`], color=blue):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");@
qu.2.1.uid=0cc1442d-d120-4b18-a4c7-b81d37f56426@
qu.2.1.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.1.weighting=1@
qu.2.1.numbering=alpha@
qu.2.1.part.1.name=sro_id_1@
qu.2.1.part.1.editing=useHTML@
qu.2.1.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><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$Distract</mi></mrow></mstyle></math>@
qu.2.1.part.1.fixed=@
qu.2.1.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2.575</mn></mrow></mstyle></math>@
qu.2.1.part.1.question=null@
qu.2.1.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$True</mi></mrow></mstyle></math>@
qu.2.1.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><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Distract</mi></mrow></mstyle></math>@
qu.2.1.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$True</mi></mrow></mstyle></math>@
qu.2.1.part.1.mode=Multiple Choice@
qu.2.1.part.1.display=vertical@
qu.2.1.part.1.answer=3@
qu.2.1.question=<p>Suppose the null hypothesis of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>&nbsp;is being tested against&nbsp;the one-sided alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&gt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>, at the 1% level of significance.&nbsp; If a random sample of size $n is taken, and the population is assumed to be normally distributed, what is the appropriate rejection region?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.2.mode=Inline@
qu.2.2.name=Determine Confidence Level@
qu.2.2.comment=<p>The confidence level is determined as the area under the <em>t</em> distribution, with $df degrees of freedom, between the values of -$QTLRound and $QTLRound.&nbsp; Using computer software, or approximating with a standard normal table, we can find this value to be $ConfDecimal.</p>
<p>Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Therefore, this is a $ConfLevel % confidence interval.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$df=range(11,15);
$n=$df+1;
$k=rint(3);
$ConfLevel=switch($k, 95, 90, 99);
$ConfDecimal=$ConfLevel/100;
$Tail=switch($k, 0.975, 0.95, 0.995);
$QTLRaw=invstudentst($df, $Tail);
$QTLRound=numfmt("0.000", $QTLRaw);
$p=plotmaple("
f := Statistics[PDF](StudentT($df),x): 
p1 := plot(f, x=-3..-1*$QTLRound, colour=blue): 
p2 := plot(f, x=-1*$QTLRound..$QTLRound, colour=blue, filled = true): 
p3 := plot(f, x=$QTLRound..3, colour=blue):
p4 := plots[textplot]([-1*$QTLRound, -0.05, `-$QTLRound`], color=blue):
p5 := plots[textplot]([$QTLRound, -0.05, `$QTLRound`], color=blue):
p6 := plots[textplot]([0.1, 0.14, `<-$ConfDecimal->`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6}), plotoptions='width=350,height=350'
");@
qu.2.2.uid=21dd7c7f-2808-4e4c-a5ca-9b9c4dea8f0f@
qu.2.2.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.2.weighting=1@
qu.2.2.numbering=alpha@
qu.2.2.part.1.name=sro_id_1@
qu.2.2.part.1.answer.units=@
qu.2.2.part.1.numStyle=   @
qu.2.2.part.1.editing=useHTML@
qu.2.2.part.1.showUnits=false@
qu.2.2.part.1.err=0.01@
qu.2.2.part.1.question=(Unset)@
qu.2.2.part.1.mode=Numeric@
qu.2.2.part.1.grading=toler_abs@
qu.2.2.part.1.negStyle=both@
qu.2.2.part.1.answer.num=$ConfLevel@
qu.2.2.question=<p>Suppose a random sample of size $n is selected from a normally distributed population.&nbsp; If the confidence interval for the population mean is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plusmn;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$QTLRound</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>s</mi></mrow><mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow><mrow></mrow></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, then what is the corresponding confidence level?</p><p>&nbsp;</p><p>Express your answer as a percent, but do NOT include the percent sign (%) in your response.</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.3.mode=Inline@
qu.2.3.name=Estimate range of one-sided p-value (1)@
qu.2.3.comment=<p>a)&nbsp; The alternative hypothesis indicates a one-sided, lower-tailed test.&nbsp; Therefore, the p-value is the area to the left of the test statistic,&nbsp;under a <em>t</em> distribution with <em>$DF</em> degrees of freedom.&nbsp; Graphically, this becomes:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Using computer software or a <em>t</em> distribution table, we can find this area to be <em>$pvalue</em>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Because the p-value is extremely large (close to 1.0), there is no significant&nbsp;evidence at any of the 10%, 5% or 1% levels of significance against the null hypothesis.</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$n1=range(7,11);
$n2=range(17, 21);
$df1=$n1-1;
$df2=$n2-1;
$tTest1=rand(3.8, 4.3, 4);
$tTest2=rand(3.0, 3.5, 4);
$k=rint(2);
$N=switch($k, $n1, $n2);
$DF=switch($k, $df1, $df2);
$tStat=switch($k, $tTest1, $tTest2);
$pvalue=studentst($DF, $tStat);
condition:gt($pvalue,0.50);
$p=plotmaple("
f := Statistics[PDF](StudentT($DF),x): 
p1 := plot(f, x=-5..$tStat, colour=blue, filled=true): 
p2 := plot(f, x=$tStat..5, colour=blue): 
p3 := plots[textplot]([$tStat, -0.05, `$tStat`], color=blue):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");@
qu.2.3.uid=09c6d0d6-fb1f-4368-96c9-694cb045759d@
qu.2.3.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.3.weighting=1,1@
qu.2.3.numbering=alpha@
qu.2.3.part.1.name=sro_id_1@
qu.2.3.part.1.editing=useHTML@
qu.2.3.part.1.choice.5=p-value < 0.005@
qu.2.3.part.1.fixed=@
qu.2.3.part.1.choice.4=0.005 < p-value < 0.025@
qu.2.3.part.1.question=null@
qu.2.3.part.1.choice.3=0.025 < p-value < 0.01@
qu.2.3.part.1.choice.2=0.10 < p-value < 0.50@
qu.2.3.part.1.choice.1=p-value > 0.50@
qu.2.3.part.1.mode=Non Permuting Multiple Choice@
qu.2.3.part.1.display=vertical@
qu.2.3.part.1.answer=1@
qu.2.3.part.2.grader=exact@
qu.2.3.part.2.name=sro_id_2@
qu.2.3.part.2.editing=useHTML@
qu.2.3.part.2.display.permute=true@
qu.2.3.part.2.question=(Unset)@
qu.2.3.part.2.answer.2=No@
qu.2.3.part.2.answer.1=Yes@
qu.2.3.part.2.mode=List@
qu.2.3.part.2.display=menu@
qu.2.3.part.2.credit.2=1.0@
qu.2.3.part.2.credit.1=0.0@
qu.2.3.question=<p>Suppose the null hypothesis of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>&nbsp;is being tested against&nbsp;the one-sided alternative hypothesis&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><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&lt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mstyle></math> in a population that is assumed to be normally distributed.&nbsp; If a random sample of size $N is taken, and the&nbsp;<em>t </em>&nbsp;test statistic is calculated to be <em>t</em> = $tStat, then:</p><p>&nbsp;</p><p>a)&nbsp; The p-value falls within which one of the following ranges:</p><p>&nbsp;</p><p><1></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Is there a significant amount of evidence against the null hypothesis at each of the 10%, 5% and 1% levels of significance?</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.2.4.mode=Inline@
qu.2.4.name=Determine one-sided rejection region (2)@
qu.2.4.comment=<p>For a one-sided, lower-tailed test, the rejection region is determined by the <em>t</em>&nbsp;value that has an area to the left, under&nbsp;a <em>t&nbsp;</em>distribution with $df degrees of freedom, of&nbsp;0.05.&nbsp; Using computer software or a <em>t</em> distribution table, we can find this value to be $True.&nbsp; Graphically, the rejection region is seen as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$n=range(5,10);
$df=$n-1;
$TrueRaw=invstudentst($df, 0.05);
$True=numfmt("#.000", $TrueRaw);
$DistractRaw=invstudentst($df, 0.025);
$Distract=numfmt("#.000", $DistractRaw);
$p=plotmaple("
f := Statistics[PDF](StudentT($df),x): 
p1 := plot(f, x=-3..$True, colour=blue, filled=true): 
p2 := plot(f, x=$True..3, colour=blue): 
p3 := plots[textplot]([$True, -0.05, `$True`], color=blue):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");@
qu.2.4.uid=d9048373-8c9e-4547-919c-667461df7c8a@
qu.2.4.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.4.weighting=1@
qu.2.4.numbering=alpha@
qu.2.4.part.1.name=sro_id_1@
qu.2.4.part.1.editing=useHTML@
qu.2.4.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><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$Distract</mi></mrow></mstyle></math>@
qu.2.4.part.1.fixed=@
qu.2.4.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1.96</mn></mrow></mstyle></math>@
qu.2.4.part.1.question=null@
qu.2.4.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$True</mi></mrow></mstyle></math>@
qu.2.4.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>t</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>$Distract</mi></mrow></mstyle></math>@
qu.2.4.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$True</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>@
qu.2.4.part.1.mode=Multiple Choice@
qu.2.4.part.1.display=vertical@
qu.2.4.part.1.answer=1@
qu.2.4.question=<p>Suppose the null hypothesis of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>&nbsp;is being tested against&nbsp;the one-sided alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&lt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, at the 5% level of significance.&nbsp; If a random sample of size $n is taken, and the population is assumed to be normally distributed, what is the appropriate rejection region?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.5.mode=Inline@
qu.2.5.name=Calculate one-sided test statistic, p-value, conclusion (1)@
qu.2.5.comment=<p>a)&nbsp; The <em>t</em> test statistic is calculated with the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</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><mfrac><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&mu;</mi></mrow></mfenced><mrow><mfrac><mi>s</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, 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><mi>t</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><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn></mrow></mfenced><mrow><mfrac><msqrt><mrow><mn>6.5</mn></mrow></msqrt><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tStat</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The p-value is calculated as the area to the left of the test statistic, under a <em>t</em> distribution with&nbsp;<em>$n - 1 = $df</em> degrees of freedom, as the alternative hypothesis indicates we are conducting a one-sided, lower-tailed test.&nbsp; Using computer software or a <em>t</em> distribution table, the p-value is found to be <em>$pvalue</em>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value = $pvalue is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis at the 10% level of significance.</p>@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=$n=range(15, 20);
$df=$n-1;
$xbar=rand(5.1, 5.5, 3);
$tStat=($xbar - 5)/(sqrt(6.5)/sqrt($n));
$pvalue=studentst($df, $tStat);
condition:gt($pvalue,0.10);@
qu.2.5.uid=9e5a31b5-3418-40a0-bbcc-ab47132bba66@
qu.2.5.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.5.weighting=1,1,1@
qu.2.5.numbering=alpha@
qu.2.5.part.1.name=sro_id_1@
qu.2.5.part.1.answer.units=@
qu.2.5.part.1.numStyle=   @
qu.2.5.part.1.editing=useHTML@
qu.2.5.part.1.showUnits=false@
qu.2.5.part.1.err=0.0010@
qu.2.5.part.1.question=(Unset)@
qu.2.5.part.1.mode=Numeric@
qu.2.5.part.1.grading=toler_abs@
qu.2.5.part.1.negStyle=both@
qu.2.5.part.1.answer.num=$tStat@
qu.2.5.part.2.name=sro_id_2@
qu.2.5.part.2.editing=useHTML@
qu.2.5.part.2.choice.5=p-value < 0.01@
qu.2.5.part.2.fixed=@
qu.2.5.part.2.choice.4=0.01 < p-value < 0.025@
qu.2.5.part.2.question=null@
qu.2.5.part.2.choice.3=0.025 < p-value < 0.05@
qu.2.5.part.2.choice.2=0.05 < p-value < 0.10@
qu.2.5.part.2.choice.1=p-value > 0.10@
qu.2.5.part.2.mode=Non Permuting Multiple Choice@
qu.2.5.part.2.display=vertical@
qu.2.5.part.2.answer=1@
qu.2.5.part.3.grader=exact@
qu.2.5.part.3.name=sro_id_3@
qu.2.5.part.3.editing=useHTML@
qu.2.5.part.3.display.permute=true@
qu.2.5.part.3.question=(Unset)@
qu.2.5.part.3.answer.2=No@
qu.2.5.part.3.answer.1=Yes@
qu.2.5.part.3.mode=List@
qu.2.5.part.3.display=menu@
qu.2.5.part.3.credit.2=1.0@
qu.2.5.part.3.credit.1=0.0@
qu.2.5.question=<p>Suppose a random sample of size $n is taken from a normally distributed population, and the sample mean and variance are calculated to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math>&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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>6.5</mn></mrow></mstyle></math>respectively.</p><p>Use this information to test the null hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>5</mn></mrow></mstyle></math>versus the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&lt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>5</mn></mrow></mstyle></math>, at the 10% level of significance.</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the test statistic, for testing the null hypothesis that the population mean is equal to 5?</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; The p-value falls within which one of the following ranges:</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Is the null hypothesis rejected at the 10% level of significance?</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.2.6.mode=Inline@
qu.2.6.name=Determine margin of error for 90%, 95% confidence interval@
qu.2.6.comment=<p>a)&nbsp; To determine the margin of error, we first need to determine the&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><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>value.&nbsp; For a <em>t</em> distribution with $df degrees of freedom, for a 95% confidence interval for the mean this value is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$t95</mi></mrow></mstyle></math>.&nbsp; Therefore, the margin of error for a 95% confidence interval for the population mean 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>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mi>s</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$t95</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi>$sigma</mi></mrow><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME95</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; For a 90% confidence interval for the population mean, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$t90</mi></mrow></mstyle></math>, and therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mi>s</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$t90</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi>$sigma</mi></mrow><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME90</mi></mrow></mstyle></math>.</p>@
qu.2.6.editing=useHTML@
qu.2.6.solution=@
qu.2.6.algorithm=$n=range(10,20);
$df=$n-1;
$sigma=rand(5,7,2);
$t95=invstudentst($df, 0.975);
$t90=invstudentst($df, 0.95);
$ME95=$t95*($sigma/sqrt($n));
$ME90=$t90*($sigma/sqrt($n));@
qu.2.6.uid=4d45f9fe-62c4-4f26-978d-568779ab1c79@
qu.2.6.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.6.weighting=1,1@
qu.2.6.numbering=alpha@
qu.2.6.part.1.name=sro_id_1@
qu.2.6.part.1.answer.units=@
qu.2.6.part.1.numStyle=   @
qu.2.6.part.1.editing=useHTML@
qu.2.6.part.1.showUnits=false@
qu.2.6.part.1.err=0.0010@
qu.2.6.part.1.question=(Unset)@
qu.2.6.part.1.mode=Numeric@
qu.2.6.part.1.grading=toler_abs@
qu.2.6.part.1.negStyle=both@
qu.2.6.part.1.answer.num=$ME95@
qu.2.6.part.2.name=sro_id_2@
qu.2.6.part.2.answer.units=@
qu.2.6.part.2.numStyle=   @
qu.2.6.part.2.editing=useHTML@
qu.2.6.part.2.showUnits=false@
qu.2.6.part.2.err=0.0010@
qu.2.6.part.2.question=(Unset)@
qu.2.6.part.2.mode=Numeric@
qu.2.6.part.2.grading=toler_abs@
qu.2.6.part.2.negStyle=both@
qu.2.6.part.2.answer.num=$ME90@
qu.2.6.question=<p>Suppose a random sample of size $n was taken from a normally distributed population, and the sample standard deviation was calculated to be <em>s</em> = $sigma.</p><p>&nbsp;</p><p>a)&nbsp; Calculate the margin of error for a 95% confidence interval for the population mean.</p><p>&nbsp;</p><p>Round your response to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Calculate the margin of error for a 90% confidence interval for the population mean.</span></p><p>&nbsp;</p><p><span>Round your response to at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.2.7.mode=Inline@
qu.2.7.name=Estimate range of two-sided p-value@
qu.2.7.comment=<p>a)&nbsp; The alternative hypothesis indicates a two-sided test, and therefore the p-value is given 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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tStat</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>, where&nbsp;<em>t</em>&nbsp;follows a <em>t </em>distribution with $DF degrees of freedom.&nbsp; Graphically, this becomes:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Using computer software or a <em>t</em> distribution table, the p-value is found to be <em>$pvalue</em>.</p>
<p>&nbsp;</p>
<p>&nbsp;b)&nbsp; Since the p-value = $pvalue is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis at the 5% level of significance.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value =$pvalue is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.01</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis at the 5% level of significance.</p>@
qu.2.7.editing=useHTML@
qu.2.7.solution=@
qu.2.7.algorithm=$n1=range(25,30);
$n2=range(10, 15);
$df1=$n1-1;
$df2=$n2-1;
$tTest1=rand(2.1, 2.4, 4);
$tTest2=rand(2.3, 2.6, 4);
$k=rint(2);
$N=switch($k, $n1, $n2);
$DF=switch($k, $df1, $df2);
$tStat=switch($k, $tTest1, $tTest2);
$pvalue=2*(1-studentst($DF, $tStat));
condition:lt($pvalue,0.05);
condition:gt($pvalue,0.02);
$p=plotmaple("
f := Statistics[PDF](StudentT($DF),x): 
p1 := plot(f, x=-3..-1*$tStat, colour=blue, filled=true): 
p2 := plot(f, x=-1*$tStat..$tStat, colour=blue):
p3 := plot(f, x=$tStat..3, colour=blue, filled=true): 
p4 := plots[textplot]([-1*$tStat, -0.05, `-$tStat`], color=blue):
p5 := plots[textplot]([$tStat, -0.05, `$tStat`], color=blue):
plots[display]({p1,p2,p3,p4,p5}), plotoptions='width=350,height=350'
");@
qu.2.7.uid=8e104fd2-656a-4234-ab64-1057cd6dacf1@
qu.2.7.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.7.weighting=1,1,1@
qu.2.7.numbering=alpha@
qu.2.7.part.1.name=sro_id_1@
qu.2.7.part.1.editing=useHTML@
qu.2.7.part.1.choice.5=p-value < 0.01@
qu.2.7.part.1.fixed=@
qu.2.7.part.1.choice.4=0.01 < p-value < 0.02@
qu.2.7.part.1.question=null@
qu.2.7.part.1.choice.3=0.02 < p-value < 0.05@
qu.2.7.part.1.choice.2=0.05 < p-value < 0.10@
qu.2.7.part.1.choice.1=p-value > 0.10@
qu.2.7.part.1.mode=Non Permuting Multiple Choice@
qu.2.7.part.1.display=vertical@
qu.2.7.part.1.answer=3@
qu.2.7.part.2.grader=exact@
qu.2.7.part.2.name=sro_id_2@
qu.2.7.part.2.editing=useHTML@
qu.2.7.part.2.display.permute=true@
qu.2.7.part.2.question=(Unset)@
qu.2.7.part.2.answer.2=No@
qu.2.7.part.2.answer.1=Yes@
qu.2.7.part.2.mode=List@
qu.2.7.part.2.display=menu@
qu.2.7.part.2.credit.2=0.0@
qu.2.7.part.2.credit.1=1.0@
qu.2.7.part.3.grader=exact@
qu.2.7.part.3.name=sro_id_3@
qu.2.7.part.3.editing=useHTML@
qu.2.7.part.3.display.permute=true@
qu.2.7.part.3.question=(Unset)@
qu.2.7.part.3.answer.2=No@
qu.2.7.part.3.answer.1=Yes@
qu.2.7.part.3.mode=List@
qu.2.7.part.3.display=menu@
qu.2.7.part.3.credit.2=1.0@
qu.2.7.part.3.credit.1=0.0@
qu.2.7.question=<p>Suppose the null hypothesis of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>&nbsp;is being tested against&nbsp;the two-sided alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&ne;</mo></mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></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> in a population that is assumed to be normally distributed.&nbsp; If a random sample of size $N is taken, and the&nbsp;<em>t </em>&nbsp;test statistic is calculated to be <em>t</em> = $tStat, then:</p><p>&nbsp;</p><p>a)&nbsp; The p-value falls within which one of the following ranges:</p><p>&nbsp;</p><p><1></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Is the null hypothesis rejected at the 5% level of significance?</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>c)&nbsp; Is the null hypothesis rejected at the 1% level of significance?</span></p><p><span><span>&nbsp;</span><3><span>&nbsp;</span>&nbsp;</span></p>@

qu.2.8.mode=Inline@
qu.2.8.name=Determine t_alpha/2@
qu.2.8.comment=<p>The <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>value&nbsp;for a $ConfLevel % confidence interval is from a <em>t </em>distribution with $df degrees of freedom, such that the area between +/- <em>t</em> is equal to $ConfDecimal.</p>
<p>Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Using computer software, or approximating with a <em>t</em> distribution table, we can find the value of <em>t</em>&nbsp; to be <em>t = $QTLRound</em>.</p>@
qu.2.8.editing=useHTML@
qu.2.8.solution=@
qu.2.8.algorithm=$df=range(16,29);
$k=rint(3);
$ConfLevel=switch($k, 95, 90, 99);
$ConfDecimal=$ConfLevel/100;
$Tail=switch($k, 0.975, 0.95, 0.995);
$QTLRaw=invstudentst($df, $Tail);
$QTLRound=numfmt("0.000", $QTLRaw);
$p=plotmaple("
f := Statistics[PDF](StudentT($df),x): 
p1 := plot(f, x=-3..-1*$QTLRound, colour=blue):
p2 := plot(f, x=-1*$QTLRound..$QTLRound, colour=blue, filled=true): 
p3 := plot(f, x=$QTLRound..3, colour=blue): 
p4 := plots[textplot]([-1*$QTLRound, -0.05, `-t`], color=blue):
p5 := plots[textplot]([$QTLRound, -0.05, `t`], color=blue):
p6 := plots[textplot]([0.1, 0.14, `$ConfDecimal`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6}), plotoptions='width=350,height=350'
");@
qu.2.8.uid=4a3c6e38-a75d-479c-a592-18962986cd9a@
qu.2.8.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.2.8.question=<p>What is the appropriate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>for a $ConfLevel % confidence interval with $df degrees of freedom?</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.9.mode=Inline@
qu.2.9.name=Calculate one-sided test statistic, p-value, conclusion (2)@
qu.2.9.comment=<p>a)&nbsp; To calculate the <em>t</em> test statistic, we use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</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><mfrac><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&mu;</mi></mrow></mfenced><mrow><mfrac><mi>s</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, 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><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn></mrow></mfenced><mrow><mfrac><msqrt><mrow><mn>0.5</mn></mrow></msqrt><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tStat</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates we are conducting a one-sided, upper-tailed test.&nbsp; Therefore, the p-value is the area to the right of the test statistic, under the <em>t</em> distribution with&nbsp;<em>$n - 1 =$df</em>.&nbsp; Using computer software or a <em>t</em> distribution table, we can find the p-value to be <em>$pvalue</em>.</p>
<p>&nbsp;</p>
<p>c) i)&nbsp; The p-value = $pvalue is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&alpha;</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><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, therefore there is sufficient evidence to reject the null hypothesis at the 5% level of significance.</p>
<p>&nbsp;</p>
<p>ii)&nbsp; The p-value = $pvalue is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.01</mn></mrow></mstyle></math>, therefore there is insufficient evidence to reject the null hypothesis at the 1% level of significance.</p>@
qu.2.9.editing=useHTML@
qu.2.9.solution=@
qu.2.9.algorithm=$n=range(21, 24);
$df=$n-1;
$xbar=rand(5.2, 5.4, 3);
$tStat=($xbar - 5)/(sqrt(0.5)/sqrt($n));
$Tail=studentst($df, $tStat);
$pvalue=1-$Tail;
condition:gt($pvalue,0.025);
condition:lt($pvalue,0.05);@
qu.2.9.uid=d138b5e4-1389-4440-9cec-3060accbf448@
qu.2.9.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.2.9.part.2.editing=useHTML@
qu.2.9.part.2.choice.5=p-value < 0.01@
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qu.2.9.part.2.choice.4=0.01 < p-value < 0.025@
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qu.2.9.question=<p>Suppose a random sample of size $n is taken from a normally distributed population, and the sample mean and variance are calculated to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math>&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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.5</mn></mrow></mstyle></math>respectively.</p><p>Use this information to test the null hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>5</mn></mrow></mstyle></math>&nbsp;versus the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&gt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>5</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the test statistic, for testing the null hypothesis that the population mean is equal to 5?</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; The p-value falls within which one of the following ranges:</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; i)&nbsp; Is the null hypothesis rejected at the 5% level of significance?</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p><span><span><span>ii)&nbsp; Is the null hypothesis rejected at the 1% level of significance?</span></span></span></p><p><span><span><span><span>&nbsp;</span><4><span>&nbsp;</span></span></span></span></p>@

qu.2.10.mode=Inline@
qu.2.10.name=Estimate range of one-sided p-value (2)@
qu.2.10.comment=<p>a)&nbsp; Since the alternative hypothesis indicates a one-sided, upper-tailed test, the p-value is the area to the right of the test statistic, under the <em>t</em> distribution&nbsp;with $DF degrees of freedom.&nbsp; Graphically, this becomes:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Using computer software or a <em>t</em> distribution table, we can find this area to be $pvalue.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>b)&nbsp; Since the p-value is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis, at the 5% level of significance.</p>@
qu.2.10.editing=useHTML@
qu.2.10.solution=@
qu.2.10.algorithm=$n1=range(7,11);
$n2=range(17, 21);
$df1=$n1-1;
$df2=$n2-1;
$tTest1=rand(2.0, 2.2, 4);
$tTest2=rand(1.8, 2.0, 4);
$k=rint(2);
$N=switch($k, $n1, $n2);
$DF=switch($k, $df1, $df2);
$tStat=switch($k, $tTest1, $tTest2);
$pvalue=1-studentst($DF, $tStat);
$p=plotmaple("
f := Statistics[PDF](StudentT($DF),x): 
p1 := plot(f, x=-3..$tStat, colour=blue): 
p2 := plot(f, x=$tStat..3, colour=blue, filled=true): 
p3 := plots[textplot]([$tStat, -0.05, `$tStat`], color=blue):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");
condition:lt($pvalue,0.05);
condition:gt($pvalue,0.025);@
qu.2.10.uid=b470e61a-9967-445a-a17e-3b49b09d8e2a@
qu.2.10.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.2.10.part.1.choice.5=p-value < 0.005@
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qu.2.10.part.1.choice.4=0.005 < p-value < 0.01@
qu.2.10.part.1.question=null@
qu.2.10.part.1.choice.3=0.01 < p-value < 0.025@
qu.2.10.part.1.choice.2=0.025 < p-value < 0.05@
qu.2.10.part.1.choice.1=p-value > 0.10@
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qu.2.10.question=<p>Suppose the null hypothesis of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>&nbsp;is being tested against&nbsp;the one-sided alternative&nbsp; hypothesis&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><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&gt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> in a population that is assumed to be normally distributed.&nbsp; If a random sample of size $N is taken, and the&nbsp;<em>t </em>&nbsp;test statistic is calculated to be <em>t</em> = $tStat, then:</p><p>&nbsp;</p><p>a)&nbsp; The p-value falls within which one of the following ranges:</p><p>&nbsp;</p><p><1></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Is there a significant amount of evidence against the null hypothesis&nbsp;at the 5% level of significance?</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.2.11.mode=Inline@
qu.2.11.name=Calculate two-sided test statistic, p-value, conclusion@
qu.2.11.comment=<p>a)&nbsp; To calculate the <em>t</em>&nbsp; test statistic, we use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>t</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><mfrac><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><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>&mu;</mi></mrow></mfenced><mrow><mfrac><mi>s</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, 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><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>50</mn></mrow></mfenced><mrow><mfrac><msqrt><mrow><mn>4.5</mn></mrow></msqrt><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tStat</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are performing a two-sided test.&nbsp; Therefore, the p-value is calculated as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tStat</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>, where <em>t</em> follows a <em>t</em> distribution with <em>$n - 1 = $df </em>degrees of freedom.&nbsp; Using computer software or a <em>t</em> distribution table, the area in the tail is $Tail, resulting in a p-value of 2 x $Tail = $pvalue.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Because the p-value = $pvalue is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis at the 5% level of significance.&nbsp;</p>@
qu.2.11.editing=useHTML@
qu.2.11.solution=@
qu.2.11.algorithm=$n=range(8, 12);
$df=$n-1;
$xbar=rand(46, 48, 3);
$tStat=($xbar - 50)/(sqrt(4.5)/sqrt($n));
$Tail=studentst($df, $tStat);
$pvalue=2*$Tail;
condition:lt($pvalue,0.005);@
qu.2.11.uid=63b27cea-7537-42db-b39c-637eb28ff335@
qu.2.11.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.11.weighting=1,1,1@
qu.2.11.numbering=alpha@
qu.2.11.part.1.name=sro_id_1@
qu.2.11.part.1.answer.units=@
qu.2.11.part.1.numStyle=   @
qu.2.11.part.1.editing=useHTML@
qu.2.11.part.1.showUnits=false@
qu.2.11.part.1.err=0.0010@
qu.2.11.part.1.question=(Unset)@
qu.2.11.part.1.mode=Numeric@
qu.2.11.part.1.grading=toler_abs@
qu.2.11.part.1.negStyle=both@
qu.2.11.part.1.answer.num=$tStat@
qu.2.11.part.2.name=sro_id_2@
qu.2.11.part.2.editing=useHTML@
qu.2.11.part.2.choice.5=p-value < 0.01@
qu.2.11.part.2.fixed=@
qu.2.11.part.2.choice.4=0.01 < p-value < 0.025@
qu.2.11.part.2.question=null@
qu.2.11.part.2.choice.3=0.025 < p-value < 0.05@
qu.2.11.part.2.choice.2=0.05 < p-value < 0.10@
qu.2.11.part.2.choice.1=p-value > 0.10@
qu.2.11.part.2.mode=Non Permuting Multiple Choice@
qu.2.11.part.2.display=vertical@
qu.2.11.part.2.answer=5@
qu.2.11.part.3.grader=exact@
qu.2.11.part.3.name=sro_id_3@
qu.2.11.part.3.editing=useHTML@
qu.2.11.part.3.display.permute=true@
qu.2.11.part.3.question=(Unset)@
qu.2.11.part.3.answer.2=No@
qu.2.11.part.3.answer.1=Yes@
qu.2.11.part.3.mode=List@
qu.2.11.part.3.display=menu@
qu.2.11.part.3.credit.2=0.0@
qu.2.11.part.3.credit.1=1.0@
qu.2.11.question=<p>Suppose a random sample of size $n is taken from a normally distributed population, and the sample mean and variance are calculated to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math>&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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>4.5</mn></mrow></mstyle></math>respectively.</p><p>Use this information to test the null hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>50</mn></mrow></mstyle></math>versus the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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'>&ne;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>50</mn></mrow></mstyle></math>, at the 5% level of significance.</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the test statistic, for testing the null hypothesis that the population mean is equal to 50?</p><p>&nbsp;</p><p>Round your response to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; The p-value falls within which one of the following ranges:</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Is the null hypothesis rejected at the 5% level of significance?</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.2.12.mode=Inline@
qu.2.12.name=Determine two-sided rejection region@
qu.2.12.comment=<p>For a two-sided hypothesis test, the rejection region is determined by a value of <em>t</em> such that for a <em>t</em> distribution with $df degrees of freedom, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>T</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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><mfrac><mn>0.10</mn><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.05</mn></mrow></mstyle></math>.&nbsp; Using computer software, or approximating with a <em>t</em> distribution table, we can find the value of&nbsp;<em>T</em>&nbsp; to be $TrueRaw.&nbsp; Therefore, the rejection region is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$True</mi></mrow></mstyle></math>, which is represented graphically as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.2.12.editing=useHTML@
qu.2.12.solution=@
qu.2.12.algorithm=$n=range(20,25);
$df=$n-1;
$TrueRaw=invstudentst($df, 0.95);
$True=numfmt("#.000", $TrueRaw);
$DistractRaw=invstudentst($df, 0.90);
$Distract=numfmt("#.000", $DistractRaw);
$p=plotmaple("
f := Statistics[PDF](StudentT($df),x): 
p1 := plot(f, x=-3..-1*$TrueRaw, colour=blue, filled=true): 
p2 := plot(f, x=-1*$TrueRaw..$TrueRaw, colour=blue): 
p3 := plot(f, x=$TrueRaw..3, colour=blue, filled=true):
p4 := plots[textplot]([-1*$TrueRaw, -0.05, `-$True`], color=blue):
p5 := plots[textplot]([$TrueRaw, -0.05, `$True`], color=blue):
plots[display]({p1,p2,p3,p4,p5}), plotoptions='width=350,height=350'
");@
qu.2.12.uid=93f2a2ee-0539-42ee-9787-24bda5cf6f1c@
qu.2.12.info=  Course=Introductory Statistics;
  Topic=Population Standard Deviation Unknown, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.12.weighting=1@
qu.2.12.numbering=alpha@
qu.2.12.part.1.name=sro_id_1@
qu.2.12.part.1.editing=useHTML@
qu.2.12.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><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$Distract</mi></mrow></mstyle></math>@
qu.2.12.part.1.fixed=@
qu.2.12.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>1.645</mn></mrow></mstyle></math>@
qu.2.12.part.1.question=null@
qu.2.12.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>$True</mi></mrow></mstyle></math>@
qu.2.12.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><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Distract</mi></mrow></mstyle></math>@
qu.2.12.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$True</mi></mrow></mstyle></math>@
qu.2.12.part.1.mode=Multiple Choice@
qu.2.12.part.1.display=vertical@
qu.2.12.part.1.answer=1@
qu.2.12.question=<p>Suppose the null hypothesis of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</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><mrow><msub><mi>&mu;</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mstyle></math>&nbsp;is being tested against a two-sided alternative hypothesis, at the 10% level of significance.&nbsp; If a random sample of size $n is taken, and the population is assumed to be normally distributed, what is the appropriate rejection region?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

