qu.1.topic=Continuous Random Variables@

qu.1.1.mode=Inline@
qu.1.1.name=Triangle Continuous Distribution: Calculate height, P(X > x)@
qu.1.1.comment=<p>a)&nbsp; To find the height of the triangle at it's highest point, we can make use of the fact that for any density function, the total area under the curve must be equal to 1.&nbsp; The area for a triangle is given by 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>Area</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>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>; since we know the base is equal to $Upper - $Lower = 20, and the area must be equal to 1, we can find the height by rearranging the equation, such that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Area</mi></mrow><mrow><mi>Base</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>1</mn></mrow><mrow><mn>20</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced></mrow></mstyle></math>, we need to determine the area under the density curve to the right of $Value, as seen in the plot below:</p>
<p>&nbsp;</p>
<p align="center"><div align="center">
	<applet width="630" height="387" code="applets.labelImage.LabelImage" archive="modules/applets.jar">
		<param name="image" value="__BASE_URI__Pictures/TriangleDistribution_Soln.jpg" />
		<param name="size" value="5" />
		<param name="label.1.x" value="345" />
		<param name="label.1.y" value="210" />
		<param name="label.1.text" value="h = $Height" />
		<param name="label.2.x" value="230" />
		<param name="label.2.y" value="320" />
		<param name="label.2.text" value="$Value" />    
		<param name="label.3.x" value="315" />
		<param name="label.3.y" value="320" />
		<param name="label.3.text" value="$MidPoint" />
		<param name="label.4.x" value="140" />
		<param name="label.4.y" value="320" />
		<param name="label.4.text" value="$Lower" />
		<param name="label.5.x" value="485" />
		<param name="label.5.y" value="320" />
		<param name="label.5.text" value="$Upper" />
	</applet>
</div>&nbsp;</p>
<p>&nbsp;</p>
<p>This area is not a standard geometric shape, however, the area to the left of $Value is a simple triangle, and since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1</mn></mrow></mstyle></math>, we can find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</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>$Value</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced></mrow></mstyle></math>, we can use the properties of similar triangles.&nbsp; The right angle triangle that forms the area to the left of $Value has a base that is $Proportion times the base of the right angle triangle that makes up half of the density curve.&nbsp; Therefore, the height of the smaller triangle is $Proportion times the height of the larger triangle.&nbsp; This results in an area of the smaller triangle that is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Area</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$SmallBase</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$SmallHeight</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Comp</mi><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>X</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>$Value</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, and therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$Comp</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Answer</mi></mrow></mstyle></math></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$LargeBase=10;
$Lower=range(10,15);
$MidPoint=$Lower + $LargeBase;
$Upper=$Lower+(2*$LargeBase);
$Height=2/($Upper-$Lower);
$Proportion=switch(rint(2), 0.2,0.4,0.5);
$SmallBase=$Proportion*10;
$SmallHeight=$Proportion*$Height;
$Value=$Lower+$SmallBase;
$Comp=($SmallBase*$SmallHeight)/2;
$Answer=1-$Comp;@
qu.1.1.uid=ab1dea8b-0913-4ce5-bef8-d61d3912ad95@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Continuous Distributions;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=Continuous distribution image - triangle;
  Type=Calculation;
@
qu.1.1.weighting=1,1@
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.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=exact_value@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=0.1@
qu.1.1.part.2.name=sro_id_2@
qu.1.1.part.2.answer.units=@
qu.1.1.part.2.numStyle=   @
qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.showUnits=false@
qu.1.1.part.2.err=0.01@
qu.1.1.part.2.question=(Unset)@
qu.1.1.part.2.mode=Numeric@
qu.1.1.part.2.grading=toler_abs@
qu.1.1.part.2.negStyle=both@
qu.1.1.part.2.answer.num=$Answer@
qu.1.1.question=<p>Let <em>X</em> be a continuous random variable,&nbsp;with the density function seen here:</p><div align="center"><applet archive="modules/applets.jar" code="applets.labelImage.LabelImage" width="630" height="387"><param name="image" value="__BASE_URI__Pictures/TriangleDistribution.jpg" /><param name="size" value="2" /><param name="label.1.x" value="140" /><param name="label.1.y" value="320" /><param name="label.1.text" value="$Lower" /><param name="label.2.x" value="485" /><param name="label.2.y" value="320" /><param name="label.2.text" value="$Upper" /></applet></div><p>&nbsp;</p><p>a)&nbsp; What is the&nbsp;height of the triangle at its highest point?</p><p>&nbsp;</p><p>Enter your response as a decimal value, NOT a fraction.</p><p><span>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>b)&nbsp; What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$Value</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round&nbsp;your response to&nbsp;at least&nbsp;2 decimal places.</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p>@

qu.2.topic=Continuous Uniform Distribution@

qu.2.1.mode=Inline@
qu.2.1.name=Determining Upper Bound@
qu.2.1.comment=<p>To find the upper bound of the uniform distribution, we can start by finding the height.&nbsp; We are told that the distribution has a lower bound of $Lower, and that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$SmallArea</mi></mrow></mstyle></math>, for $Value < <em>M</em>.&nbsp; A plot of this information results in:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Since we know the area to the left of $Value, we can use that information to find the height.&nbsp; Recall that the area of a rectangle is given by <em>Area = Base X Height</em>, which means that we can find the height be to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>Area</mi><mrow><mi>Base</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$SmallArea</mi><mrow><mi>$Value</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Lower</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.&nbsp;</p>
<p>Now that we have the height of the distribution, we can find the length of the base of the entire distribution by using the fact that the entire area under the curve must be equal to 1.&nbsp; Therefore, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>Area</mi><mrow><mi>Height</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi>$Height</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$LargeBase</mi></mrow></mstyle></math>.&nbsp; However, this is just the length of the base; to find the upper bound of the distribution, we need <em>M = $Lower + $LargeBase = $Upper</em>.&nbsp;</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$HeightOptions=[0.025, 0.04, 0.05];
$Height=switch(rint(4), $HeightOptions);
$Lower=range(10,15);
$LargeBase=1/$Height;
$Upper=$Lower+$LargeBase;
$Value=range(($Lower+5), ($Upper-5));
$SmallBase=$Value-$Lower;
$SmallArea=$SmallBase*$Height;
$p=plotmaple("
f := Statistics[PDF](Uniform($Lower, $Upper),x): 
p1 := plot(f, x=$Lower-10..$Value, colour=blue, filled=true): 
p2 := plot(f, x=$Value..$Upper+10, colour=blue): 
p3 := plots[textplot]([$Value, -0.005, `$Value`], color=blue):
p4 := plots[textplot]([$Upper, -0.005, `M`], color=blue):
p5 := plots[textplot]([$Lower+4, 0.01, `$SmallArea`], color=black):
plots[display]({p1,p2,p3,p4,p5}), plotoptions='width=350,height=350'
");@
qu.2.1.uid=48102698-c189-4fde-b25b-26b54c6b2c2a@
qu.2.1.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Hard;
  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.answer.units=@
qu.2.1.part.1.numStyle=   @
qu.2.1.part.1.editing=useHTML@
qu.2.1.part.1.showUnits=false@
qu.2.1.part.1.question=(Unset)@
qu.2.1.part.1.mode=Numeric@
qu.2.1.part.1.grading=exact_value@
qu.2.1.part.1.negStyle=both@
qu.2.1.part.1.answer.num=$Upper@
qu.2.1.question=<p>Let the random variable <em>X</em> have a continuous uniform distribution with a minimum value of $Lower and an unknown maximum value, <em>M</em>.</p><p>&nbsp;</p><p>If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$Value</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SmallArea</mi></mrow></mstyle></math>, where $Value < <em>M</em>, what is the value of <em>M</em>?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.2.mode=Inline@
qu.2.2.name=Calculate P(X > x)@
qu.2.2.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced></mrow></mstyle></math>, we need to determine the area under&nbsp;the uniform distribution to the right of $Value.&nbsp; We can visualize this area in the following plot:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Since the total area under the curve is equal to 1, the height of the distribution is found by rearranging 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>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</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>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>to be&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>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mn>1</mn><mrow><mi>Base</mi></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi>$Upper</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Lower</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.&nbsp; Now, to find the desired probability, we simply need to calculate the given area as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</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><mfenced open='(' close=')' separators=','><mrow><mi>$Upper</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Probability</mi></mrow></mstyle></math>.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=[110, 115, 120];
$b=[160, 165, 170];
$Lower=switch(rint(3), $a);
$Upper=switch(rint(3), $b);
$Value=rand($Lower+5, $Upper-5, 5);
$Height=1/($Upper-$Lower);
$Probability=($Upper-$Value)*$Height;
$p=plotmaple("
f := Statistics[PDF](Uniform($Lower, $Upper),x): 
p1 := plot(f, x=$Lower-10.0..$Value, colour=blue): 
p2 := plot(f, x=$Value..$Upper+10.0, colour=blue, filled=true): 
p3 := plots[textplot]([$Value, -0.003, `$Value`], color=blue):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");@
qu.2.2.uid=253f27fe-5907-40c4-b2b1-c362d79425eb@
qu.2.2.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.2.weighting=1@
qu.2.2.numbering=alpha@
qu.2.2.part.1.name=sro_id_1@
qu.2.2.part.1.answer.units=@
qu.2.2.part.1.numStyle=   @
qu.2.2.part.1.editing=useHTML@
qu.2.2.part.1.showUnits=false@
qu.2.2.part.1.err=0.0010@
qu.2.2.part.1.question=(Unset)@
qu.2.2.part.1.mode=Numeric@
qu.2.2.part.1.grading=toler_abs@
qu.2.2.part.1.negStyle=both@
qu.2.2.part.1.answer.num=$Probability@
qu.2.2.question=<p>Let the random variable X have a continuous uniform distribution with a minimum value of $Lower and a maximum value of $Upper.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round&nbsp;your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.3.mode=Inline@
qu.2.3.name=Calculating the Mean@
qu.2.3.comment=<p>In order to find the mean of the distribution, we can make use of the fact that the continuous uniform distribution is a symmetric distribution, and therefore the mean and the median are the same.</p>
<p>To find the median, we need a value of <em>X</em> such that the area to the left (or to the right) of <em>X</em> is equal to 0.5.&nbsp; Since we know the upper and lower bound of the distribution, we can find the height by remembering that the total area under the curve must be equal to 1.&nbsp; Therefore, 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>Height</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><mi>Area</mi><mrow><mi>Base</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi>$Upper</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Lower</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.&nbsp; Now, to find a value of <em>X</em> such that the area below it is 0.5, we can rearrange the equation for the area of a rectangle to solve for the base: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>Area</mi><mrow><mi>Height</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>0.5</mn><mrow><mi>$Height</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Base</mi></mrow></mstyle></math>.&nbsp; Since this is just the <em>length</em> of the base, we need to find the value of <em>X</em> as <em>X = $Base + $Lower = $Mean, </em>which is the value of both the median and the mean of the distribution.</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$Lower=range(40,50);
$Upper=range(80,90);
$Height=1/($Upper-$Lower);
$Base=(0.5/$Height);
$Mean=$Base + $Lower;@
qu.2.3.uid=b983697b-1edc-47dd-9539-4dffedb086e0@
qu.2.3.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.2.3.weighting=1@
qu.2.3.numbering=alpha@
qu.2.3.part.1.name=sro_id_1@
qu.2.3.part.1.answer.units=@
qu.2.3.part.1.numStyle=   @
qu.2.3.part.1.editing=useHTML@
qu.2.3.part.1.showUnits=false@
qu.2.3.part.1.err=0.01@
qu.2.3.part.1.question=(Unset)@
qu.2.3.part.1.mode=Numeric@
qu.2.3.part.1.grading=toler_abs@
qu.2.3.part.1.negStyle=both@
qu.2.3.part.1.answer.num=$Mean@
qu.2.3.question=<p>Let the random variable X have a continuous uniform distribution with a minimum value of $Lower and a maximum value of $Upper.</p><p>&nbsp;</p><p>What is the mean of the distribution?</p><p>&nbsp;</p><p>Round your response to at least 2 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.4.mode=Inline@
qu.2.4.name=Calculate P(X > x Intersect X < y)@
qu.2.4.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced></mrow></mstyle></math>, we need to determine the area under the continuous uniform distribution for values of <em>X</em> that are greater than $Value1 <strong>and</strong> less than $Value2.&nbsp; Graphically, this area is seen as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>In order to find this area, we must first find the height of the distribution.&nbsp; We can do this by making use of the fact that the total area under the curve must be equal to 1.&nbsp; Therefore, 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>Area</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>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</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><mi>Area</mi><mrow><mi>Base</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mn>150</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><mn>100</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.&nbsp; Now, to find the area between $Value1 and $Value2, we use the equation for area of a rectangle: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$Value2</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>$Value1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Height</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>$Intersect</mi></mrow></mstyle></math>.</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$Height=1/(150-100);
$Value1=rand(110, 120, 5);
$Value2=rand(130, 140, 5);
$Intersect=($Value2-$Value1)*$Height;
$p=plotmaple("
f := Statistics[PDF](Uniform(100, 150),x): 
p1 := plot(f, x=90..$Value1, colour=blue):
p2 := plot(f, x=$Value1..$Value2, colour=blue, filled=true): 
p3 := plot(f, x=$Value2..160, colour=blue): 
p4 := plots[textplot]([$Value1, -0.005, `$Value1`], color=blue):
p5 := plots[textplot]([$Value2, -0.005, `$Value2`], color=blue):
plots[display]({p1,p2,p3,p4,p5}), plotoptions='width=350,height=350'
");@
qu.2.4.uid=d1f9903b-e0ce-42f8-9320-84e72152885b@
qu.2.4.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Graph of Uniform Distribution;
  Type=Calculation;
@
qu.2.4.weighting=1@
qu.2.4.numbering=alpha@
qu.2.4.part.1.name=sro_id_1@
qu.2.4.part.1.answer.units=@
qu.2.4.part.1.numStyle=   @
qu.2.4.part.1.editing=useHTML@
qu.2.4.part.1.showUnits=false@
qu.2.4.part.1.err=0.0010@
qu.2.4.part.1.question=(Unset)@
qu.2.4.part.1.mode=Numeric@
qu.2.4.part.1.grading=toler_abs@
qu.2.4.part.1.negStyle=both@
qu.2.4.part.1.answer.num=$Intersect@
qu.2.4.question=<p>Let <em>X</em> be a continuous uniform random variable with&nbsp;the following distribution:</p><p><img alt="" width="384" height="242" src="__BASE_URI__Pictures/UniformDistribution1.jpg" /></p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&cap;</mo></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$Value2</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</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.2.5.mode=Multiple Selection@
qu.2.5.name=Definitions 2@
qu.2.5.comment=@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=@
qu.2.5.uid=a5802058-685b-454d-9c9e-a03d53b99d2d@
qu.2.5.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.5.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>Note that there may be more than one correct answer; select all that are true.</p>@
qu.2.5.answer=1, 2, 3@
qu.2.5.choice.1=In a continuous uniform distribution, defined on the interval [a, b], then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>a</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi>b</mi></mrow></mfenced></mrow></mstyle></math>.@
qu.2.5.choice.2=The mean in a continuous uniform distribution 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>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>b</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>.@
qu.2.5.choice.3=In a continuous uniform distribution defined on the interval [a,b], <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>b</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>.@
qu.2.5.choice.4=For a continuous uniform random variable, X, defined on the interval [a,b], P(X = x) is the same non-zero value for any x between a and b.@
qu.2.5.choice.5=A continuous uniform distribution is not a symmetric distribution, unless it is centered around 0.@
qu.2.5.fixed=@

qu.2.6.mode=Multiple Selection@
qu.2.6.name=Definitions 1@
qu.2.6.comment=@
qu.2.6.editing=useHTML@
qu.2.6.solution=@
qu.2.6.algorithm=@
qu.2.6.uid=6e5ac2fb-4f66-407f-b89e-76acecf89037@
qu.2.6.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.6.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>Note that there may be more than one correct answer; select all that are true.</p>@
qu.2.6.answer=1, 2@
qu.2.6.choice.1=In a continuous uniform distribution, the height of the curve, f(x), is the same for all values of the random variable X. @
qu.2.6.choice.2=In a continuous uniform distribution, the mean and the median are the same.@
qu.2.6.choice.3=There are countably infinite values of X in a continuous uniform distribution.@
qu.2.6.choice.4=For a continuous uniform distribution defined on the interval [a,b], P(X < a) and P(X > b) is undefined.@
qu.2.6.choice.5=The mean and the variance of a continuous uniform random variable are the same.@
qu.2.6.fixed=@

qu.2.7.mode=Inline@
qu.2.7.name=Calculate P(X < x U X > y)@
qu.2.7.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced></mrow></mstyle></math>, we can 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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>, which in this case becomes <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$Value1</mi></mrow></mfenced><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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp;&nbsp; Notice, however, that the lower bound of the distribution is $Lower, and that $Value2 is smaller than this value.&nbsp; We know that for a continuous uniform distribution, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Lower</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></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><mfenced open='(' close=')' separators=','><mrow><mi>X</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>$Value2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></mrow></mstyle></math>&nbsp;as well.&nbsp; Furthermore, there are no values of <em>X</em> that will be less than $Value2 <strong>and</strong> greater than $Value1, therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>, which means we can simplify our desired probability to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$Value1</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Graphically, we can see this area to be:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>To find this area, we first must determine the height of the distribution.&nbsp; To do this, we make use of the fact that the total area under the curve must be equal to 1, therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Area</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>Area</mi><mrow><mi>Base</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi>$Upper</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>$Lower</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.&nbsp; Finally, to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced></mrow></mstyle></math>, we use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</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><mfenced open='(' close=')' separators=','><mrow><mi>$Upper</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Union</mi></mrow></mstyle></math></p>@
qu.2.7.editing=useHTML@
qu.2.7.solution=@
qu.2.7.algorithm=$a=[110, 115, 120];
$b=[160, 165, 170];
$Lower=switch(rint(3), $a);
$Upper=switch(rint(3), $b);
$Value1=rand(($Lower+20), ($Upper-5), 5);
$Value2=rand(($Lower-15), ($Lower-5), 5);
$Height=1/($Upper-$Lower);
$Union=($Upper-$Value1)*$Height;
$p=plotmaple("
f := Statistics[PDF](Uniform($Lower, $Upper),x): 
p1 := plot(f, x=$Lower-20..$Value1, colour=blue): 
p2 := plot(f, x=$Value1..$Upper+10, colour=blue, filled=true): 
p3 := plots[textplot]([$Value1, -0.005, `$Value1`], color=blue):
p4 := plots[textplot]([$Value2, -0.005, `$Value2`], color=blue):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");@
qu.2.7.uid=bae99d1f-e88f-4b70-970a-c789209cd51f@
qu.2.7.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.2.7.weighting=1@
qu.2.7.numbering=alpha@
qu.2.7.part.1.name=sro_id_1@
qu.2.7.part.1.answer.units=@
qu.2.7.part.1.numStyle=   @
qu.2.7.part.1.editing=useHTML@
qu.2.7.part.1.showUnits=false@
qu.2.7.part.1.err=0.0010@
qu.2.7.part.1.question=(Unset)@
qu.2.7.part.1.mode=Numeric@
qu.2.7.part.1.grading=toler_abs@
qu.2.7.part.1.negStyle=both@
qu.2.7.part.1.answer.num=$Union@
qu.2.7.question=<p>Let the random variable X have a continuous uniform distribution with a minimum value of $Lower and a maximum value of $Upper.</p><p>&nbsp;</p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$Value2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round&nbsp;your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.8.mode=Inline@
qu.2.8.name=P(X < x U X > y) with Graph@
qu.2.8.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced></mrow></mstyle></math>, we can make use of 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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>, which in this case becomes <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Graphically, we can see <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mstyle></math>&nbsp;as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Because there are no values of <em>X</em> that are both less than $Value1 <strong>and</strong> greater than $Value2, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></mrow></mstyle></math>.&nbsp; In order to find the remaining probabilities, we first must find the height of the distribution.&nbsp; To do this, we can make use of the fact that the total area under the curve is equal to 1.&nbsp; Therefore, rearranging the equation for the area of a rectangle, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Area</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>Area</mi><mrow><mi>Base</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mn>150</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>100</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced></mrow></mstyle></math>, we need to determine the area under the continuous uniform distribution to the left of $Value1, which is simply <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Area</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$Value1</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>100</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Prob1</mi></mrow></mstyle></math>.&nbsp; Similarly, to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced></mrow></mstyle></math>, we need to determine the area under the continuous uniform&nbsp;distribution to the right of $Value2, which is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Area</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mn>150</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Prob2</mi></mrow></mstyle></math>.</p>
<p>Finally, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Prob1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Prob2</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>$Union</mi></mrow></mstyle></math>.&nbsp;</p>@
qu.2.8.editing=useHTML@
qu.2.8.solution=@
qu.2.8.algorithm=$Height=1/(150-100);
$Value1=rand(110, 120, 5);
$Value2=rand(130, 140, 5);
$Prob1=($Value1-100)*$Height;
$Prob2=(150-$Value2)*$Height;
$Union=$Prob1+$Prob2;
$p=plotmaple("
f := Statistics[PDF](Uniform(100, 150),x): 
p1 := plot(f, x=90..$Value1, colour=blue, filled=true): 
p2 := plot(f, x=$Value1..$Value2, colour=blue): 
p3 := plot(f, x=$Value2..160, colour=blue, filled=true): 
p4 := plots[textplot]([$Value1, -0.003, `$Value1`], color=blue):
p5 := plots[textplot]([$Value2, -0.003, `$Value2`], color=blue):
plots[display]({p1,p2,p3,p4,p5}), plotoptions='width=350,height=350'
");@
qu.2.8.uid=70731f2a-9586-4d57-bd12-224bdb1cefe9@
qu.2.8.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Graph of Uniform Distribution;
  Type=Calculation;
@
qu.2.8.weighting=1@
qu.2.8.numbering=alpha@
qu.2.8.part.1.name=sro_id_1@
qu.2.8.part.1.answer.units=@
qu.2.8.part.1.numStyle=   @
qu.2.8.part.1.editing=useHTML@
qu.2.8.part.1.showUnits=false@
qu.2.8.part.1.err=0.0010@
qu.2.8.part.1.question=(Unset)@
qu.2.8.part.1.mode=Numeric@
qu.2.8.part.1.grading=toler_abs@
qu.2.8.part.1.negStyle=both@
qu.2.8.part.1.answer.num=$Union@
qu.2.8.question=<p>Let <em>X</em> be a continuous uniform random variable with&nbsp;the following distribution:</p><p><img alt="" width="384" height="242" src="__BASE_URI__Pictures/UniformDistribution1.jpg" /></p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$Value2</mi></mrow></mfenced></mrow></mstyle></math>?</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 P(X > x| X < y)@
qu.2.9.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>X</mi><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue2</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>, we need to utilize the general conditional probability 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>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>A</mi><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>B</mi><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><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.&nbsp; In this case, the formula becomes <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue2</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue2</mi></mrow></mfenced></mrow><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue2</mi></mrow></mfenced></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Beginning with the numerator, to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue2</mi></mrow></mfenced></mrow></mstyle></math>, we need to find the area under the uniform curve for values of <em>X</em> that are greater than $CondValue1 <strong>and </strong>less than $CondValue2.&nbsp; Graphically, we can see this area as:</p>
<p>&nbsp;</p>
<p align="center">$P1</p>
<p>&nbsp;</p>
<p>To find this area, we can use the formula for area of a rectangle: <em>Area = Base X Height</em>.&nbsp; We can first find the height of the distribution by recalling that the total area under the distribution is equal to 1.&nbsp; 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><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</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><mfenced open='(' close=')' separators=','><mrow><mi>$Upper</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Lower</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>Height</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><mn>1</mn><mrow><mi>$Upper</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Lower</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.&nbsp; Now, to find the area between $CondValue1 and $CondValue2, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$CondValue2</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>$CondValue1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Height</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Numer</mi></mrow></mstyle></math>.</p>
<p>Now we need to determine <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue2</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; To find this value, we need to determine the area under the uniform distribution that is to the left of $CondValue2, as see in the plot below:</p>
<p>&nbsp;</p>
<p align="center">$P2</p>
<p>&nbsp;</p>
<p>To find this area, we again use the formula for area of a rectangle: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$CondValue2</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>$Lower</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Denom</mi></mrow></mstyle></math>.</p>
<p>Finally, we can find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue1</mi><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>X</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>$CondValue2</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>by dividing the two probabilities we have found: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue1</mi><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>X</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>$CondValue2</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$Numer</mi><mrow><mi>$Denom</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Prob</mi></mrow></mstyle></math></p>@
qu.2.9.editing=useHTML@
qu.2.9.solution=@
qu.2.9.algorithm=$a=[110, 115, 120];
$b=[160, 165, 170];
$Lower=switch(rint(3), $a);
$Upper=switch(rint(3), $b);
$Height=1/($Upper-$Lower);
$CondValue1=rand(($Lower+5), ($Upper-5), 5);
$CondValue2=rand(($Lower+5), ($Upper-5), 5);
condition:lt($CondValue1,($CondValue2-5));
$Numer=($CondValue2-$CondValue1)*$Height;
$Denom=($CondValue2-$Lower)*$Height;
$Prob=$Numer/$Denom;
$P1=plotmaple("
f := Statistics[PDF](Uniform($Lower, $Upper),x): 
p1 := plot(f, x=$Lower-10..$CondValue1, colour=blue): 
p2 := plot(f, x=$CondValue1..$CondValue2, colour=blue, filled=true): 
p3 := plot(f, x=$CondValue2..$Upper+10, colour=blue):
p4 := plots[textplot]([$CondValue1, -0.003, `$CondValue1`], color=blue):
p5 := plots[textplot]([$CondValue2, -0.005, `$CondValue2`], color=blue):
plots[display]({p1,p2,p3,p4,p5}), plotoptions='width=350,height=350'
");
$P2=plotmaple("
f := Statistics[PDF](Uniform($Lower, $Upper),x): 
p1 := plot(f, x=$Lower-10..$CondValue2, colour=blue, filled=true):  
p2 := plot(f, x=$CondValue2..$Upper+10, colour=blue):
p3 := plots[textplot]([$CondValue2, -0.005, `$CondValue2`], color=blue):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");@
qu.2.9.uid=2c90357e-2014-40f7-8526-4502903c183e@
qu.2.9.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.9.weighting=1@
qu.2.9.numbering=alpha@
qu.2.9.part.1.name=sro_id_1@
qu.2.9.part.1.answer.units=@
qu.2.9.part.1.numStyle=   @
qu.2.9.part.1.editing=useHTML@
qu.2.9.part.1.showUnits=false@
qu.2.9.part.1.err=0.01@
qu.2.9.part.1.question=(Unset)@
qu.2.9.part.1.mode=Numeric@
qu.2.9.part.1.grading=toler_abs@
qu.2.9.part.1.negStyle=both@
qu.2.9.part.1.answer.num=$Prob@
qu.2.9.question=<p>Let the random variable X have a continuous uniform distribution with a minimum value of $Lower and a maximum value of $Upper.</p><p>&nbsp;</p><p>What is<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$CondValue1</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$CondValue2</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round&nbsp;your response to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.10.mode=Inline@
qu.2.10.name=Definitions 1&2: Random selection of True/False@
qu.2.10.comment=@
qu.2.10.editing=useHTML@
qu.2.10.solution=@
qu.2.10.algorithm=$k1=rint(10);
$k2=rint(10);
$k3=rint(10);
$k4=rint(10);
$k5=rint(10);
$z=maple("S := $k1,$k2,$k3,$k4,$k5:
floor( nops({S})/nops([S]) )");
condition: $z;
$a=("'In a continuous uniform distribution, the height of the curve, f(x), is the same for all values of the random variable X.'");
$b=("'In a continuous uniform distribution, the mean and the median are the same.'");
$c=("'In a continuous uniform distribution defined on the interval [a,b], P(a < X < b) = 1.'");
$d=("'There are countably infinite values of X in a continuous uniform distribution.'");
$e=("'For a continuous uniform distribution defined on the interval [a,b], P(X < a) and P(X > b) is undefined.'");
$f=("'The mean and the variance of a continuous uniform random variable are the same.'");
$g=("'For a continuous uniform random variable, X, defined on the interval [a,b], P(X = x) is the same non-zero value for any x between a and b.'");
$h=("'A continuous uniform distribution is not a symmetric distribution, unless it is centered around 0.'");
$i=("'In a continuous uniform distribution, defined on the interval [a, b], the P(X < a) = P(X > b).'");
$j1=maple("
J1:=convert(cat(`The mean in a continuous uniform distribution is`,MathML[ExportPresentation](mu = (b-a)/2),`.`),string):
J1
");
$j=switch(0, $j1);
$Answers=["'True'","'True'","'True'","'False'","'False'","'False'","'False'","'False'","'True'","'True'"];
$Distractors=["'False'","'False'","'False'","'True'","'True'","'True'","'True'","'True'","'False'","'False'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A1=switch($k1, $Answers);
$D1=switch($k1, $Distractors);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A2=switch($k2, $Answers);
$D2=switch($k2, $Distractors);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A3=switch($k3, $Answers);
$D3=switch($k3, $Distractors);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A4=switch($k4, $Answers);
$D4=switch($k4, $Distractors);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A5=switch($k5, $Answers);
$D5=switch($k5, $Distractors);@
qu.2.10.uid=ee888bd0-6ad1-4a97-a4a9-53928b832559@
qu.2.10.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.2.10.weighting=1,1,1,1,1@
qu.2.10.numbering=alpha@
qu.2.10.part.1.grader=exact@
qu.2.10.part.1.name=sro_id_1@
qu.2.10.part.1.editing=useHTML@
qu.2.10.part.1.display.permute=true@
qu.2.10.part.1.question=(Unset)@
qu.2.10.part.1.answer.2=$D1@
qu.2.10.part.1.answer.1=$A1@
qu.2.10.part.1.mode=List@
qu.2.10.part.1.display=menu@
qu.2.10.part.1.credit.2=0.0@
qu.2.10.part.1.credit.1=1.0@
qu.2.10.part.2.grader=exact@
qu.2.10.part.2.name=sro_id_2@
qu.2.10.part.2.editing=useHTML@
qu.2.10.part.2.display.permute=true@
qu.2.10.part.2.question=(Unset)@
qu.2.10.part.2.answer.2=$D2@
qu.2.10.part.2.answer.1=$A2@
qu.2.10.part.2.mode=List@
qu.2.10.part.2.display=menu@
qu.2.10.part.2.credit.2=0.0@
qu.2.10.part.2.credit.1=1.0@
qu.2.10.part.3.grader=exact@
qu.2.10.part.3.name=sro_id_3@
qu.2.10.part.3.editing=useHTML@
qu.2.10.part.3.display.permute=true@
qu.2.10.part.3.question=(Unset)@
qu.2.10.part.3.answer.2=$D3@
qu.2.10.part.3.answer.1=$A3@
qu.2.10.part.3.mode=List@
qu.2.10.part.3.display=menu@
qu.2.10.part.3.credit.2=0.0@
qu.2.10.part.3.credit.1=1.0@
qu.2.10.part.4.grader=exact@
qu.2.10.part.4.name=sro_id_4@
qu.2.10.part.4.editing=useHTML@
qu.2.10.part.4.display.permute=true@
qu.2.10.part.4.question=(Unset)@
qu.2.10.part.4.answer.2=$D4@
qu.2.10.part.4.answer.1=$A4@
qu.2.10.part.4.mode=List@
qu.2.10.part.4.display=menu@
qu.2.10.part.4.credit.2=0.0@
qu.2.10.part.4.credit.1=1.0@
qu.2.10.part.5.grader=exact@
qu.2.10.part.5.name=sro_id_5@
qu.2.10.part.5.editing=useHTML@
qu.2.10.part.5.display.permute=true@
qu.2.10.part.5.question=(Unset)@
qu.2.10.part.5.answer.2=$D5@
qu.2.10.part.5.answer.1=$A5@
qu.2.10.part.5.mode=List@
qu.2.10.part.5.display=menu@
qu.2.10.part.5.credit.2=0.0@
qu.2.10.part.5.credit.1=1.0@
qu.2.10.question=<p>Identify each of the following statements as either true or false.</p><p>&nbsp;</p><p>a)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> $Q1</p><p>&nbsp;</p><p>b)&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span> $Q2</p><p>&nbsp;</p><p>c)&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span> $Q3</p><p>&nbsp;</p><p>d)&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span> $Q4</p><p>&nbsp;</p><p>e)&nbsp;<span>&nbsp;</span><5><span>&nbsp; $Q5</span></p>@

qu.2.11.mode=Inline@
qu.2.11.name=Calculate Percentile@
qu.2.11.comment=<p>To calculate the $Percentile<sup>$Superscript</sup> of the uniform distribution, we need to find a value of <em>X</em> such that the area to the left of <em>X</em>, under the given uniform distribution, will be&nbsp;equal to $Area.&nbsp; Graphically, we can see this as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>To find <em>X</em>, we can make use of the equation for area of a rectangle: <em>Area = Base X Height</em>.&nbsp; Since the total area under the curve must be 1, we can find the height of the distribution&nbsp;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>Height</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><mn>1</mn><mrow><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi>$Upper</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Lower</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.&nbsp; Now, to find the base of the&nbsp;shaded region, we make&nbsp;use of the fact that the&nbsp;shaded area is known to be 0.90.&nbsp; Therefore, the base&nbsp;can be found by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>Area</mi><mrow><mi>Height</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$Area</mi><mrow><mi>$Height</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Base</mi></mrow></mstyle></math>.&nbsp; However, this is just the<em> length</em> of the base; to&nbsp;find the actual value of<em> X</em>, we need<em> X = $Base + $Lower = $Answer</em>&nbsp;</p>@
qu.2.11.editing=useHTML@
qu.2.11.solution=@
qu.2.11.algorithm=$a=[115, 120, 125];
$b=[160, 165, 170];
$Lower=switch(rint(3), $a);
$Upper=switch(rint(3), $b);
$Height=1/($Upper-$Lower);
$Percentile=range(60,90);
$Area=$Percentile/100;
$Base=($Area/$Height);
$Answer=$Base + $Lower;
$ScriptOptions=["'th'","'st'","'nd'","'rd'","'th'","'th'","'th'","'th'","'th'","'th'"];
$Remain=maple("irem($Percentile, 10)");
$Superscript=switch($Remain,$ScriptOptions);
$p=plotmaple("
f := Statistics[PDF](Uniform($Lower, $Upper),x): 
p1 := plot(f, x=$Lower-5..$Answer, colour=blue, filled=true): 
p2 := plot(f, x=$Answer..$Upper+5, colour=blue): 
p3 := plots[textplot]([$Answer, -0.003, `X`], color=blue):
p4 := plots[textplot]([$Answer-10, 0.01, `<-$Area->`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");@
qu.2.11.uid=820d96db-756f-40ee-b174-81eb00769fbe@
qu.2.11.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.11.weighting=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=0.01@
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=$Answer@
qu.2.11.question=<p>Let <em>X</em> have a continuous uniform distribution with a minimum value of $Lower and a maximum value of $Upper.</p><p>&nbsp;</p><p>What is the $Percentile<sup>$Superscript</sup> percentile?</p><p>&nbsp;</p><p>Round&nbsp;your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.12.mode=Inline@
qu.2.12.name=Calculate P(X < x U X < y)@
qu.2.12.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced></mrow></mstyle></math>, we can 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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>B</mi></mrow></mfenced><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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi></mrow></mfenced></mrow></mstyle></math>, which in this case becomes <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Notice, however, that <em>X < $Value1</em> is a subset of <em>X < $Value2</em>, which means that for values of <em>X</em> to be less than both <em>$Value1 <strong>and</strong> $Value2</em>, they must be strictly less than <em>$Value1</em>.&nbsp; This means that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cap;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced></mrow></mstyle></math>, and our equation above for the union of the two events simplifies to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$Value1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Value2</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Therefore, we only need to find the area under the continous uniform distribution to the left of $Value2.&nbsp; To do this, we must first find the height of the distribution, which we can find by making use of the fact that the total area under the curve must be equal to 1.&nbsp; This gives us <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Area</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>Area</mi><mrow><mi>Base</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mn>150</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>100</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Height</mi></mrow></mstyle></math>.&nbsp; We can now 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><mfenced open='(' close=')' separators=','><mrow><mi>X</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>$Value2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>Base</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$Value2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>100</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Height</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Union</mi></mrow></mstyle></math>.</p>@
qu.2.12.editing=useHTML@
qu.2.12.solution=@
qu.2.12.algorithm=$Height=1/(150-100);
$Value1=rand(110, 120, 5);
$Value2=rand(130, 140, 5);
$Union=($Value2-100)*$Height;@
qu.2.12.uid=b775248e-cf4d-470d-a85f-739fcbd0df1f@
qu.2.12.info=  Course=Introductory Statistics;
  Topic=Continuous Uniform Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=Graph of Uniform Distribution;
  Type=Calculation;
@
qu.2.12.weighting=1@
qu.2.12.numbering=alpha@
qu.2.12.part.1.name=sro_id_1@
qu.2.12.part.1.answer.units=@
qu.2.12.part.1.numStyle=   @
qu.2.12.part.1.editing=useHTML@
qu.2.12.part.1.showUnits=false@
qu.2.12.part.1.err=0.0010@
qu.2.12.part.1.question=(Unset)@
qu.2.12.part.1.mode=Numeric@
qu.2.12.part.1.grading=toler_abs@
qu.2.12.part.1.negStyle=both@
qu.2.12.part.1.answer.num=$Union@
qu.2.12.question=<p>Let <em>X</em> be a continuous uniform random variable with&nbsp;the following distribution:</p><p><img alt="" width="384" height="242" src="__BASE_URI__Pictures/UniformDistribution1.jpg" /></p><p>What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$Value1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$Value2</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>?</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.3.topic=Normal Distribution@

qu.3.1.mode=Inline@
qu.3.1.name=Calculate Percentile of X, 2 parts@
qu.3.1.comment=<p>a)&nbsp; To calculate the $Value1<sup>$Script1</sup> percentile of the distribution of <em>X</em>, we can start by finding the $Value1<sup>$Script1</sup> percentile of the standard normal distribution.&nbsp; To find the $Value1<sup>$Script1</sup> percentile of the standard normal distribution, we need to determine a&nbsp;value of <em>z</em> such that the area to the left of <em>z</em>, under the standard normal curve, is equal to $Value1Dec.&nbsp; Using computer software, or approximating with a standard normal table, the $Value1<sup>$Script1</sup> percentile of the standard normal distribution is found to be $PTL1.</p>
<p>Now that we have the $Value1<sup>$Script1</sup> percentile of the standard normal distribution, we need to find the corresponding value on the distribution of <em>X</em>, which follows a normal distribution with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</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><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>25</mn></mrow></mstyle></math>.&nbsp; We can do this by rearranging 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><mi>X</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><mi>&sigma;</mi></mrow></mfrac></mrow></mstyle></math>&nbsp;to become <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&sdot;</mo><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>X</mi></mrow></mstyle></math>; substituting in the appropriate values 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>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$PTL1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>25</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mean</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$X1</mi></mrow></mstyle></math>.&nbsp; Therefore, the $Value1<sup>$Script1</sup> percentile of the distribution of <em>X</em> is $X1.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To find the $Value2<sup>$Script2</sup> percentile of the distribution of <em>X</em>, we follow the same procedure as in part (a).&nbsp; First, to find the $Value2<sup>$Script2</sup> percentile of the standard normal distribution, we require a value of <em>z</em> such that the area to the left of <em>z</em>, under the standard normal curve, is $Value2Dec.&nbsp; Using computer software, or approximating with a standard normal table, we can find this value to be $PTL2.</p>
<p>To find the corresponding value on the distribution of <em>X</em>, we substitute the appropriate values into the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>z</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&mu;</mi></mrow></mrow></mstyle></math>, which gives us <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$PTL2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>25</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mean</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$X2</mi></mrow></mstyle></math>.&nbsp; Therefore, the $Value2<sup>$Script2</sup> percentile of the distribution of <em>X</em> is $X2.</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$mean=range(100,110);
$Value1=range(80,90);
$Value1Dec=$Value1/100;
$Value2=range(30,40);
$Value2Dec=$Value2/100;
$PTL1=inverf($Value1Dec);
$PTL2=inverf($Value2Dec);
$X1=($PTL1*25) + $mean;
$X2=($PTL2*25) + $mean;
$Data=maple("
REM1:=irem($Value1,10):
REM2:=irem($Value2,10):
REM1, REM2
");
$Remain1=switch(0, $Data);
$Remain2=switch(1, $Data);
$ScriptOpts=["'th'","'st'","'nd'","'rd'","'th'","'th'","'th'","'th'","'th'","'th'","'th'"];
$Script1=switch($Remain1, $ScriptOpts);
$Script2=switch($Remain2, $ScriptOpts);@
qu.3.1.uid=59076c7f-7c3a-4e76-9414-bef50e0fa2c3@
qu.3.1.info=  Course=Introductory Statistics;
  Topic=Normal Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.3.1.weighting=1,1@
qu.3.1.numbering=alpha@
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qu.3.1.part.1.numStyle=   @
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.showUnits=false@
qu.3.1.part.1.err=1.0@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.mode=Numeric@
qu.3.1.part.1.grading=toler_abs@
qu.3.1.part.1.negStyle=both@
qu.3.1.part.1.answer.num=$X1@
qu.3.1.part.2.name=sro_id_2@
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qu.3.1.part.2.numStyle=   @
qu.3.1.part.2.editing=useHTML@
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qu.3.1.part.2.err=1.0@
qu.3.1.part.2.question=(Unset)@
qu.3.1.part.2.mode=Numeric@
qu.3.1.part.2.grading=toler_abs@
qu.3.1.part.2.negStyle=both@
qu.3.1.part.2.answer.num=$X2@
qu.3.1.question=<p>Suppose the random variable <em>X</em> follows a normal distribution with mean <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</mi></mrow></mstyle></math>and standard deviation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mn mathvariant='italic'>25</mn></mrow></mstyle></math>.</p><p>&nbsp;</p><p>Calculate each of the following.&nbsp;</p><p>&nbsp;</p><p>In each case, round your response to&nbsp;at least&nbsp;2 decimal places.</p><p>&nbsp;</p><p>a)&nbsp; Calculate the $Value1<sup>$Script1</sup> percentile of the distribution of <em>X</em>.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Calculate the $Value2<sup>$Script2</sup> percentile of the distribution of <em>X</em>.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.3.2.mode=Inline@
qu.3.2.name=Calculate P(X >/< x), 3 parts.@
qu.3.2.comment=<p>a)&nbsp; If the random variable <em>X</em> follows a normal distribution, with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mu</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>10</mn></mrow></mstyle></math>, then we can find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xa</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;by standardizing <em>X</em>, and finding the corresponding area under a standard normal curve.</p>
<p>To standardize <em>X</em>, we can 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><mrow><mi>X</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><mi>&sigma;</mi></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we find 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>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>$xa</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mu</mi></mrow><mrow><mn>10</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$za</mi></mrow></mstyle></math>, and we can now determine <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xa</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;by calculating <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$za</mi></mrow></mfenced></mrow></mstyle></math>, where <em>Z</em> follows a standard normal distribution.&nbsp; Using computer software, or approximating with a standard normal table, we find that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$za</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$xa</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Probza</mi></mrow></mstyle></math>, as seen in the graph below:</p>
<p>&nbsp;</p>
<p align="center">$P1</p>
<p>&nbsp;</p>
<p>b)&nbsp; We can follow the same procedure as seen in part (a), where we can find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xb</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;by standardizing the random variable <em>X</em> to a standard normal random variable, <em>Z</em>.&nbsp; 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><mrow><mi>X</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><mi>&sigma;</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$xb</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$mu</mi></mrow><mrow><mn>10</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$zb</mi></mrow></mstyle></math>, the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xb</mi></mrow></mfenced></mrow></mstyle></math>is equivalent 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>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><mi>$zb</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; This area can be found through computer software, or approximating with a standard normal table.&nbsp; As seen in the plot below, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$zb</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xb</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Probzb</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p align="center">$P2</p>
<p>&nbsp;</p>
<p>c)&nbsp; We can follow the same procedure as seen in the previous sections, where we can find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>$xc1</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>X</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>$xc2</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;by standardizing the random variable <em>X</em> to the standard normal random variable <em>Z</em>.&nbsp; In this case, however, we need to use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>X</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi></mrow><mrow><mi>&sigma;</mi></mrow></mfrac></mrow></mstyle></math>twice, so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>$xc1</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>$mu</mi></mrow><mrow><mn>10</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$zc1</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><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>$xc2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$mu</mi></mrow><mrow><mn>10</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$zc2</mi></mrow></mstyle></math>.&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>$xc1</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>X</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>$xc2</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;is equivalent to <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>$zc1</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>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>$zc2</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; This area is graphically represented in the plot below, and can be found with computer software or approximating with a standard normal table.&nbsp;</p>
<p>&nbsp;</p>
<p align="center">$P3</p>
<p>&nbsp;</p>
<p>In this case,&nbsp;the area&nbsp;is found 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>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>$zc1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$zc2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>$xc1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xc2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Probzc</mi></mrow></mstyle></math></p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$mu=range(50,55);
$xa=range(40,45);
$xb=range(60,65);
$xc1=$mu-5;
$xc2=$mu+15;
$za=($xa-$mu)/10;
$zb=($xb-$mu)/10;
$zc1=($xc1-$mu)/10;
$zc2=($xc2-$mu)/10;
$Probza=erf($za);
$ProbzaDisp=decimal(4, $Probza);
$Probzb=1-erf($zb);
$ProbzbDisp=decimal(4, $Probzb);
$Probzc=erf($zc2)-$erf($zc1);
$ProbzcDisp=decimal(4, $Probzc);
$P1=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$za, colour=blue, filled=true): 
p2 := plot(f, x=$za..3, colour=blue): 
p3 := plots[textplot]([$za, -0.08, `$za`], color=blue):
p4 := plots[textplot]([-2.0, 0.04, `$ProbzaDisp`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=250,height=250'
");
$P2=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$zb, colour=blue): 
p2 := plot(f, x=$zb..3, colour=blue, filled=true): 
p3 := plots[textplot]([$zb, -0.08, `$zb`], color=blue):
p4 := plots[textplot]([2.0, 0.04, `$ProbzbDisp`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=250,height=250'
");
$P3=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$zc1, colour=blue):
p2 := plot(f, x=$zc1..$zc2, colour=blue, filled=true): 
p3 := plot(f, x=$zc2..3, colour=blue): 
p4 := plots[textplot]([$zc1, -0.08, `$zc1`], color=blue):
p5 := plots[textplot]([$zc2, -0.08, `$zc2`], color=blue):
p6 := plots[textplot]([0.18, 0.14, `$ProbzcDisp`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6}), plotoptions='width=250,height=250'
");@
qu.3.2.uid=5e7ff0f4-b0a8-4e5d-b5d6-e10c20e50806@
qu.3.2.info=  Course=Introductory Statistics;
  Topic=Normal Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.3.2.weighting=1,1,1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.answer.units=@
qu.3.2.part.1.numStyle=   @
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.showUnits=false@
qu.3.2.part.1.err=0.0010@
qu.3.2.part.1.question=(Unset)@
qu.3.2.part.1.mode=Numeric@
qu.3.2.part.1.grading=toler_abs@
qu.3.2.part.1.negStyle=both@
qu.3.2.part.1.answer.num=$Probza@
qu.3.2.part.2.name=sro_id_2@
qu.3.2.part.2.answer.units=@
qu.3.2.part.2.numStyle=   @
qu.3.2.part.2.editing=useHTML@
qu.3.2.part.2.showUnits=false@
qu.3.2.part.2.err=0.0010@
qu.3.2.part.2.question=(Unset)@
qu.3.2.part.2.mode=Numeric@
qu.3.2.part.2.grading=toler_abs@
qu.3.2.part.2.negStyle=both@
qu.3.2.part.2.answer.num=$Probzb@
qu.3.2.part.3.name=sro_id_3@
qu.3.2.part.3.answer.units=@
qu.3.2.part.3.numStyle=   @
qu.3.2.part.3.editing=useHTML@
qu.3.2.part.3.showUnits=false@
qu.3.2.part.3.err=0.0010@
qu.3.2.part.3.question=(Unset)@
qu.3.2.part.3.mode=Numeric@
qu.3.2.part.3.grading=toler_abs@
qu.3.2.part.3.negStyle=both@
qu.3.2.part.3.answer.num=$Probzc@
qu.3.2.question=<p>Suppose the random variable <em>X</em> follows a normal distribution with mean <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mu</mi></mrow></mstyle></math>and standard deviation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mn mathvariant='italic'>10</mn></mrow></mstyle></math>.</p><p>&nbsp;</p><p>Calculate each of the following.&nbsp;</p><p>&nbsp;</p><p>In each case, round your response to&nbsp;at least&nbsp;4 decimal places.</p><p>&nbsp;</p><p>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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xa</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span>b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xb</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p><span><span>c) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>$xc1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$xc2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.3.3.mode=Inline@
qu.3.3.name=Calculate P(Z >/< z), 3 parts@
qu.3.3.comment=<p>a)&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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>$z1</mi></mrow></mfenced></mrow></mstyle></math>, we need to find the area under the standard normal curve to the left of $z1.&nbsp; Using computer software, or approximating with a standard normal table, we can find this area to be $Probz1.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$P1</p>
<p>&nbsp;</p>
<p>b)&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><mi>$z2</mi></mrow></mfenced></mrow></mstyle></math>, we need to find the area under a standard normal curve to the right of $z2.&nbsp; Using computer software, or approximating with a standard normal table, we can find this value to be $Probz2.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$P2</p>
<p>&nbsp;</p>
<p>c)&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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>$z3</mi></mrow></mfenced></mrow></mstyle></math>, we need to find the area under a standard normal curve to the left of $z3.&nbsp; Using computer software, or approximating with a standard normal table, we can find this area to be $Probz3.&nbsp; Graphically, we can represent this as:</p>
<p>&nbsp;</p>
<p align="center">$P3</p>@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$z1=rand(1.5, 2.0, 3);
$z2=rand(-1.0, -0.5, 2);
$z3=rand(-0.5, 0.5, 2);
$Probz1=erf($z1);
$Probz1Disp=decimal(4, $Probz1);
$Probz2=1-erf($z2);
$Probz2Disp=decimal(4, $Probz2);
$Probz3=erf($z3);
$Probz3Disp=decimal(4, $Probz3);
$P1=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$z1, colour=blue, filled=true): 
p2 := plot(f, x=$z1..3, colour=blue): 
p3 := plots[textplot]([$z1, -0.08, `$z1`], color=blue):
p4 := plots[textplot]([0.1, 0.14, `-$Probz1Disp-`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=250,height=250'
");
$P2=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$z2, colour=blue): 
p2 := plot(f, x=$z2..3, colour=blue, filled=true): 
p3 := plots[textplot]([$z2, -0.08, `$z2`], color=blue):
p4 := plots[textplot]([0.4, 0.14, `-$Probz2Disp-`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=250,height=250'
");
$P3=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$z3, colour=blue, filled=true): 
p2 := plot(f, x=$z3..3, colour=blue): 
p3 := plots[textplot]([$z3, -0.08, `$z3`], color=blue):
p4 := plots[textplot]([-1.0, 0.14, `-$Probz3Disp-`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=250,height=250'
");@
qu.3.3.uid=3fe1b664-66d2-40ce-a588-0eaa06cb40f7@
qu.3.3.info=  Course=Introductory Statistics;
  Topic=Standard Normal Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.3.3.weighting=1,1,1@
qu.3.3.numbering=alpha@
qu.3.3.part.1.name=sro_id_1@
qu.3.3.part.1.answer.units=@
qu.3.3.part.1.numStyle=   @
qu.3.3.part.1.editing=useHTML@
qu.3.3.part.1.showUnits=false@
qu.3.3.part.1.err=0.0010@
qu.3.3.part.1.question=(Unset)@
qu.3.3.part.1.mode=Numeric@
qu.3.3.part.1.grading=toler_abs@
qu.3.3.part.1.negStyle=both@
qu.3.3.part.1.answer.num=$Probz1@
qu.3.3.part.2.name=sro_id_2@
qu.3.3.part.2.answer.units=@
qu.3.3.part.2.numStyle=   @
qu.3.3.part.2.editing=useHTML@
qu.3.3.part.2.showUnits=false@
qu.3.3.part.2.err=0.0010@
qu.3.3.part.2.question=(Unset)@
qu.3.3.part.2.mode=Numeric@
qu.3.3.part.2.grading=toler_abs@
qu.3.3.part.2.negStyle=both@
qu.3.3.part.2.answer.num=$Probz2@
qu.3.3.part.3.name=sro_id_3@
qu.3.3.part.3.answer.units=@
qu.3.3.part.3.numStyle=   @
qu.3.3.part.3.editing=useHTML@
qu.3.3.part.3.showUnits=false@
qu.3.3.part.3.err=0.0010@
qu.3.3.part.3.question=(Unset)@
qu.3.3.part.3.mode=Numeric@
qu.3.3.part.3.grading=toler_abs@
qu.3.3.part.3.negStyle=both@
qu.3.3.part.3.answer.num=$Probz3@
qu.3.3.question=<p>Suppose the random variable <em>Z</em> follows a standard normal distribution.</p><p>&nbsp;</p><p>Calculate each of the following.&nbsp; Round your answers to at least&nbsp;4 decimal places.</p><p>&nbsp;</p><p>a) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span>b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p><span><span>c) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z3</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><3><span>&nbsp;</span></span></span></p><p>&nbsp;</p>@

qu.3.4.mode=Multiple Selection@
qu.3.4.name=Definitions 2: Normal Distribution@
qu.3.4.comment=@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=@
qu.3.4.uid=8fc84b32-6e8a-43d2-ad84-8ed81c0b5a4e@
qu.3.4.info=  Course=Introductory Statistics;
  Topic=Normal Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.3.4.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.3.4.answer=1, 2@
qu.3.4.choice.1=If a continuous random variable has a normal distribution with 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> and standard deviation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>, it can be transformed to a standard normal distribution 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>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&mu;</mi></mrow></mfenced><mrow><mi>&sigma;</mi></mrow></mfrac></mrow></mstyle></math>.@
qu.3.4.choice.2=The median in a standard normal distribution is 0.@
qu.3.4.choice.3=For a standard normal random variable, Z, P(Z < -z) < P(Z > z), where z is some constant.@
qu.3.4.choice.4=Both the normal and standard normal distributions are always symmetric around 0.@
qu.3.4.choice.5=The 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> of a normal distribution must be a value greater than 0.@
qu.3.4.fixed=@

qu.3.5.mode=Inline@
qu.3.5.name=Calculate P(z1 < Z < z2), 3 parts@
qu.3.5.comment=<p>a)&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$z1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z1</mi></mrow></mfenced></mrow></mstyle></math>, we need to find the area under the standard normal curve between the values of -$z1 and $z1.&nbsp; Using computer software, or approximating with a standard normal table, we can find this value to be $Probz1.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$P1</p>
<p>&nbsp;</p>
<p>b)&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>$z2A</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z2B</mi></mrow></mfenced></mrow></mstyle></math>, we need to determine the area under the standard normal distribution in between the values of $z2A and $z2B.&nbsp; Using computer software, or approximating with a standard normal table, we can find this value to be $Probz2.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$P2</p>
<p>&nbsp;</p>
<p>c)&nbsp; To find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfenced open='|' close='|' 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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z3</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;is equivalent to finding <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$z3</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z3</mi></mrow></mfenced></mrow></mstyle></math>.&nbsp; Since these areas will be equal, we simply need to find one, and multiply that value be 2.&nbsp; Using computer software, or approximating with a standard normal distribution, we can find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$z3</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;to be $Probz3Tail.&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z3</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>is <em>2 X $Probz3Tail = $Probz3</em>.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$P3</p>@
qu.3.5.editing=useHTML@
qu.3.5.solution=@
qu.3.5.algorithm=$z1=rand(0.4, 0.9, 2);
$z2A=rand(-0.5, -0.1, 2);
$z2B=rand(0.5, 1.0, 2);
$z3=rand(1.0, 1.5, 3);
$Probz1=erf($z1)-erf(-1*$z1);
$Probz1Disp=decimal(4, $Probz1);
$Probz2=erf($z2B)-erf($z2A);
$Probz2Disp=decimal(4, $Probz2);
$Probz3Tail=erf(-1*$z3);
$Probz3TailDis=decimal(4,$Probz3Tail);
$Probz3=2*$Probz3Tail;
$Probz3Disp=decimal(4, $Probz3);
$P1=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..-1*$z1, colour=blue):
p2 := plot(f, x=-1*$z1..$z1, colour=blue, filled=true): 
p3 := plot(f, x=$z1..3, colour=blue): 
p4 := plots[textplot]([-1*$z1, -0.08, `-$z1`], color=blue):
p5 := plots[textplot]([$z1, -0.12, `$z1`], color=blue):
p6 := plots[textplot]([0.1, 0.14, `$Probz1Disp`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6}), plotoptions='width=250,height=250'
");
$P2=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$z2A, colour=blue):
p2 := plot(f, x=$z2A..$z2B, colour=blue, filled=true): 
p3 := plot(f, x=$z2B..3, colour=blue): 
p4 := plots[textplot]([$z2A, -0.08, `$z2A`], color=blue):
p5 := plots[textplot]([$z2B, -0.12, `$z2B`], color=blue):
p6 := plots[textplot]([0.1, 0.14, `$Probz2Disp`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6}), plotoptions='width=250,height=250'
");
$P3=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..-1*$z3, colour=blue, filled=true):
p2 := plot(f, x=-1*$z3..$z3, colour=blue): 
p3 := plot(f, x=$z3..3, colour=blue, filled=true): 
p4 := plots[textplot]([-1*$z3, -0.08, `-$z3`], color=blue):
p5 := plots[textplot]([$z3, -0.08, `$z3`], color=blue):
p6 := plots[textplot]([2.0, 0.08, `$Probz3TailDis`], color=black):
p7 := plots[textplot]([-2.0, 0.08, `$Probz3TailDis`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6,p7}), plotoptions='width=250,height=250'
");@
qu.3.5.uid=e791feb1-391b-466c-a02f-fd5b4d3debc0@
qu.3.5.info=  Course=Introductory Statistics;
  Topic=Standard Normal Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.3.5.weighting=1,1,1@
qu.3.5.numbering=alpha@
qu.3.5.part.1.name=sro_id_1@
qu.3.5.part.1.answer.units=@
qu.3.5.part.1.numStyle=   @
qu.3.5.part.1.editing=useHTML@
qu.3.5.part.1.showUnits=false@
qu.3.5.part.1.err=0.0010@
qu.3.5.part.1.question=(Unset)@
qu.3.5.part.1.mode=Numeric@
qu.3.5.part.1.grading=toler_abs@
qu.3.5.part.1.negStyle=both@
qu.3.5.part.1.answer.num=$Probz1@
qu.3.5.part.2.name=sro_id_2@
qu.3.5.part.2.answer.units=@
qu.3.5.part.2.numStyle=   @
qu.3.5.part.2.editing=useHTML@
qu.3.5.part.2.showUnits=false@
qu.3.5.part.2.err=0.0010@
qu.3.5.part.2.question=(Unset)@
qu.3.5.part.2.mode=Numeric@
qu.3.5.part.2.grading=toler_abs@
qu.3.5.part.2.negStyle=both@
qu.3.5.part.2.answer.num=$Probz2@
qu.3.5.part.3.name=sro_id_3@
qu.3.5.part.3.answer.units=@
qu.3.5.part.3.numStyle=   @
qu.3.5.part.3.editing=useHTML@
qu.3.5.part.3.showUnits=false@
qu.3.5.part.3.err=0.0010@
qu.3.5.part.3.question=(Unset)@
qu.3.5.part.3.mode=Numeric@
qu.3.5.part.3.grading=toler_abs@
qu.3.5.part.3.negStyle=both@
qu.3.5.part.3.answer.num=$Probz3@
qu.3.5.question=<p>Suppose the random variable <em>Z</em> follows a standard normal distribution.</p><p>&nbsp;</p><p>Calculate each of the following.&nbsp; Round your responses to at least&nbsp;4 decimal places.</p><p>&nbsp;</p><p>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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$z1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$z1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>$z2A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$z2B</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;&nbsp;</p><p><span><span><span>c)&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mfenced open='|' close='|' 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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$z3</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p>@

qu.3.6.mode=Inline@
qu.3.6.name=Calculate percentile of the standard normal distribution@
qu.3.6.comment=<p>To find the $PTL<sup>$Script</sup> percentile of the standard normal distribution, we need to find the <em>z</em> value such that the area to the left of this value, under the standard normal curve, is equal to $PTLDecimal.&nbsp; Using computer software, or approximating with a standard normal table, the <em>z</em> value with an area below it of $PTLDecimal is $ZValue.&nbsp; We can see this graphically as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.3.6.editing=useHTML@
qu.3.6.solution=@
qu.3.6.algorithm=$PTL=range(20,40);
$PTLDecimal=$PTL/100;
$ZValue=inverf($PTLDecimal);
$ZValueDisp=decimal(4, $ZValue);
$Super=maple("
Remain:=irem($PTL, 10):
Remain
");
$Endings=["'th'","'st'","'nd'","'rd'","'th'","'th'","'th'","'th'","'th'","'th'","'th'"];
$Script=switch($Super,$Endings);
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$ZValue, colour=blue, filled=true): 
p2 := plot(f, x=$ZValue..3, colour=blue): 
p3 := plots[textplot]([$ZValue, -0.05, `$ZValueDisp`], color=blue):
p4 := plots[textplot]([-1.50, 0.08, `$PTLDecimal`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");@
qu.3.6.uid=286e2292-cae7-4e46-aabf-9249a2aa0e67@
qu.3.6.info=  Course=Introductory Statistics;
  Topic=Standard Normal Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.3.6.weighting=1@
qu.3.6.numbering=alpha@
qu.3.6.part.1.name=sro_id_1@
qu.3.6.part.1.answer.units=@
qu.3.6.part.1.numStyle=   @
qu.3.6.part.1.editing=useHTML@
qu.3.6.part.1.showUnits=false@
qu.3.6.part.1.err=0.01@
qu.3.6.part.1.question=(Unset)@
qu.3.6.part.1.mode=Numeric@
qu.3.6.part.1.grading=toler_abs@
qu.3.6.part.1.negStyle=both@
qu.3.6.part.1.answer.num=$ZValue@
qu.3.6.question=<p>Calculate the $PTL<sup>$Script</sup> percentile of the standard normal distribution.</p><p>&nbsp;</p><p>Round your response to&nbsp;at least&nbsp;2 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.3.7.mode=Inline@
qu.3.7.name=Definitions 1&2: Random selection of True/False@
qu.3.7.comment=@
qu.3.7.editing=useHTML@
qu.3.7.solution=@
qu.3.7.algorithm=$k1=rint(10);
$k2=rint(10);
$k3=rint(10);
$k4=rint(10);
$k5=rint(10);
$z=maple("S := $k1,$k2,$k3,$k4,$k5:
floor( nops({S})/nops([S]) )");
condition: $z;
$a=("'The are two parameters in the normal distribution.'");
$b=("'In a standard normal distribution, the variance and the standard deviation are the same value.'");
$c=("'The median in a standard normal distribution is 0.'");
$d=("'In a normal distribution, the mean does not equal the median, but they are the same in a standard normal distribution.'");
$e=("'In a normal distribution, as the variance increases the distribution becomes narrower.'");
$f=("'For a standard normal random variable, Z, P(Z < -z) < P(Z > z), where z is some constant.'");
$g=("'Both the normal and standard normal distributions are always symmetric around 0.'");
$hij=maple("
H1:=convert(cat(`The domain of a normal distribution is `,MathML[ExportPresentation](-infinity),`to`,MathML[ExportPresentation](infinity), `.`),string):
A1:=convert(cat(`If a continuous random variable has a normal distribution with mean`,MathML[ExportPresentation](mu),`and standard deviation`,MathML[ExportPresentation](sigma),`, it can be transformed to a standard normal distribution with the formula `,MathML[ExportPresentation](z=(X-mu)/sigma),`.`),string):
J1:=convert(cat(`The mean`,MathML[ExportPresentation](mu),`of a normal distribution must be a value greater than 0.`),string):
H1, A1, J1
");
$h=switch(0, $hij);
$i=switch(1, $hij);
$j=switch(2, $hij);
$Answers=["'True'","'True'","'True'","'False'","'False'","'False'","'False'","'True'","'True'","'False'"];
$Distractors=["'False'","'False'","'False'","'True'","'True'","'True'","'True'","'False'","'False'","'True'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$g,"$h","$i","$j");
$A1=switch($k1, $Answers);
$D1=switch($k1, $Distractors);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$g,"$h","$i","$j");
$A2=switch($k2, $Answers);
$D2=switch($k2, $Distractors);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$g,"$h","$i","$j");
$A3=switch($k3, $Answers);
$D3=switch($k3, $Distractors);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$g,"$h","$i","$j");
$A4=switch($k4, $Answers);
$D4=switch($k4, $Distractors);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$g,"$h","$i","$j");
$A5=switch($k5, $Answers);
$D5=switch($k5, $Distractors);@
qu.3.7.uid=5db16959-fbb8-4d2d-85c1-985a1201ddd4@
qu.3.7.info=  Course=Introductory Statistics;
  Topic=Normal Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.3.7.weighting=1,1,1,1,1@
qu.3.7.numbering=alpha@
qu.3.7.part.1.grader=exact@
qu.3.7.part.1.name=sro_id_1@
qu.3.7.part.1.editing=useHTML@
qu.3.7.part.1.display.permute=true@
qu.3.7.part.1.question=(Unset)@
qu.3.7.part.1.answer.2=$D1@
qu.3.7.part.1.answer.1=$A1@
qu.3.7.part.1.mode=List@
qu.3.7.part.1.display=menu@
qu.3.7.part.1.credit.2=0.0@
qu.3.7.part.1.credit.1=1.0@
qu.3.7.part.2.grader=exact@
qu.3.7.part.2.name=sro_id_2@
qu.3.7.part.2.editing=useHTML@
qu.3.7.part.2.display.permute=true@
qu.3.7.part.2.question=(Unset)@
qu.3.7.part.2.answer.2=$D2@
qu.3.7.part.2.answer.1=$A2@
qu.3.7.part.2.mode=List@
qu.3.7.part.2.display=menu@
qu.3.7.part.2.credit.2=0.0@
qu.3.7.part.2.credit.1=1.0@
qu.3.7.part.3.grader=exact@
qu.3.7.part.3.name=sro_id_3@
qu.3.7.part.3.editing=useHTML@
qu.3.7.part.3.display.permute=true@
qu.3.7.part.3.question=(Unset)@
qu.3.7.part.3.answer.2=$D3@
qu.3.7.part.3.answer.1=$A3@
qu.3.7.part.3.mode=List@
qu.3.7.part.3.display=menu@
qu.3.7.part.3.credit.2=0.0@
qu.3.7.part.3.credit.1=1.0@
qu.3.7.part.4.grader=exact@
qu.3.7.part.4.name=sro_id_4@
qu.3.7.part.4.editing=useHTML@
qu.3.7.part.4.display.permute=true@
qu.3.7.part.4.question=(Unset)@
qu.3.7.part.4.answer.2=$D4@
qu.3.7.part.4.answer.1=$A4@
qu.3.7.part.4.mode=List@
qu.3.7.part.4.display=menu@
qu.3.7.part.4.credit.2=0.0@
qu.3.7.part.4.credit.1=1.0@
qu.3.7.part.5.grader=exact@
qu.3.7.part.5.name=sro_id_5@
qu.3.7.part.5.editing=useHTML@
qu.3.7.part.5.display.permute=true@
qu.3.7.part.5.question=(Unset)@
qu.3.7.part.5.answer.2=$D5@
qu.3.7.part.5.answer.1=$A5@
qu.3.7.part.5.mode=List@
qu.3.7.part.5.display=menu@
qu.3.7.part.5.credit.2=0.0@
qu.3.7.part.5.credit.1=1.0@
qu.3.7.question=<p>Identify each of the following statements as either true or false.</p><p>&nbsp;</p><p>a)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> $Q1</p><p>&nbsp;</p><p>b)&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span> $Q2</p><p>&nbsp;</p><p>c)&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span> $Q3</p><p>&nbsp;</p><p>d)&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span> $Q4</p><p>&nbsp;</p><p>e)&nbsp;<span>&nbsp;</span><5><span>&nbsp;</span> $Q5</p>@

qu.3.8.mode=Multiple Selection@
qu.3.8.name=Definitions 1: Normal Distribution@
qu.3.8.comment=@
qu.3.8.editing=useHTML@
qu.3.8.solution=@
qu.3.8.algorithm=@
qu.3.8.uid=e29199d3-af08-48bb-8fc7-27c5e556a9b6@
qu.3.8.info=  Course=Introductory Statistics;
  Topic=Normal Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.3.8.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.3.8.answer=1, 2, 3@
qu.3.8.choice.1=The domain of a normal distribution 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><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='bold-italic' fontweight='bold' lspace='0.0em' rspace='0.0em'>to</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&infin;</mi></mrow></mrow></mstyle></math>.@
qu.3.8.choice.2=The are two parameters in the normal distribution.@
qu.3.8.choice.3=In a standard normal distribution, the variance and the standard deviation are the same value.@
qu.3.8.choice.4=In a normal distribution, the mean does not equal the median, but they are the same in a standard normal distribution.@
qu.3.8.choice.5=In a normal distribution, as the variance increases the distribution becomes narrower.@
qu.3.8.fixed=@

qu.4.topic=t Distribution@

qu.4.1.mode=Inline@
qu.4.1.name=Estimate range of P(T > t) (1)@
qu.4.1.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tvalue</mi></mrow></mfenced></mrow></mstyle></math>, we need to find the area under a <em>t</em> distribution, with $df degrees of freedom, to the right of $tvalue.&nbsp; Using computer software, or approximating with a <em>t</em> distribution table, we can find this area to be $Prob.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$df=range(11,15);
$tvalue=rand(2.1, 2.6, 3);
$Tail=studentst($df, $tvalue);
$Prob=1-$Tail;
$ProbDisplay=decimal(4, $Prob);
condition:gt($Prob,0.010);
condition:lt($Prob,0.025);
$p=plotmaple("
f := Statistics[PDF](StudentT($df),x): 
p1 := plot(f, x=-3.5..$tvalue, colour=blue): 
p2 := plot(f, x=$tvalue..3.5, colour=blue, filled=true): 
p3 := plots[textplot]([$tvalue, -0.05, `$tvalue`], color=blue):
p4 := plots[textplot]([3.0, 0.02, `$ProbDisplay`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");@
qu.4.1.uid=621c2479-ee03-4dc9-ae9e-f2c8457948dc@
qu.4.1.info=  Course=Introductory Statistics;
  Topic=t Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.4.1.weighting=1@
qu.4.1.numbering=alpha@
qu.4.1.part.1.name=sro_id_1@
qu.4.1.part.1.editing=useHTML@
qu.4.1.part.1.choice.5=less than 0.005@
qu.4.1.part.1.fixed=@
qu.4.1.part.1.choice.4=0.005 to 0.010@
qu.4.1.part.1.question=null@
qu.4.1.part.1.choice.3=0.010 to 0.025@
qu.4.1.part.1.choice.2=0.025 to 0.05@
qu.4.1.part.1.choice.1=0.05 to 0.10@
qu.4.1.part.1.mode=Non Permuting Multiple Choice@
qu.4.1.part.1.display=vertical@
qu.4.1.part.1.answer=3@
qu.4.1.question=<p>If&nbsp;the random variable <em>T</em> follows a t distribution with $df degrees of freedom, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>T</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tvalue</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;falls within which one of the following ranges?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.4.2.mode=Multiple Selection@
qu.4.2.name=Definitions 1: t Distribution@
qu.4.2.comment=@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=@
qu.4.2.uid=de0ee47c-c8a4-4a45-8deb-1c714dc7b70c@
qu.4.2.info=  Course=Introductory Statistics;
  Topic=t Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.4.2.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.4.2.answer=1, 2, 3@
qu.4.2.choice.1=Like the normal distribution, the t distribution is symmetric, but it has heavier tails than the normal distribution.@
qu.4.2.choice.2=As the sample size increases, the t distribution approaches the standard normal distribution.@
qu.4.2.choice.3=It is appropriate to use the t distribution when dealing with small sample sizes drawn from normally distributed populations.@
qu.4.2.choice.4=The t distribution is centered around 1.@
qu.4.2.choice.5=Like the normal distribution, the t distribution depends on the parameters <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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> 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>, as well as the degrees of freedom.@
qu.4.2.fixed=@

qu.4.3.mode=Inline@
qu.4.3.name=Estimate range of P(T > t) (2)@
qu.4.3.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tvalue</mi></mrow></mfenced></mrow></mstyle></math>, we need to determine the area under a <em>t</em> distribution, with $df degrees of freedom, to the right of $tvalue.&nbsp; Using computer software, or approximating from a <em>t </em>distribution table, we can find this area to be $Prob, which is between 0.05 and 0.10.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$df=range(21,25);
$tvalue=rand(1.40, 1.70, 4);
$Tail=studentst($df, $tvalue);
$Prob=1-$Tail;
$ProbDisplay=decimal(4, $Prob);
condition:gt($Prob,0.05);
condition:lt($Prob,0.10);
$p=plotmaple("
f := Statistics[PDF](StudentT($df),x): 
p1 := plot(f, x=-3..$tvalue, colour=blue): 
p2 := plot(f, x=$tvalue..3, colour=blue, filled=true): 
p3 := plots[textplot]([$tvalue, -0.05, `$tvalue`], color=blue):
p4 := plots[textplot]([2.0, 0.02, `$ProbDisplay`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");@
qu.4.3.uid=b554ca4c-e767-4408-baca-913083e84781@
qu.4.3.info=  Course=Introductory Statistics;
  Topic=t Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.4.3.weighting=1@
qu.4.3.numbering=alpha@
qu.4.3.part.1.name=sro_id_1@
qu.4.3.part.1.editing=useHTML@
qu.4.3.part.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0.005</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>T</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$tvalue</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0.01</mn></mrow></mstyle></math>@
qu.4.3.part.1.fixed=@
qu.4.3.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0.01</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>T</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$tvalue</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0.025</mn></mrow></mstyle></math>@
qu.4.3.part.1.question=null@
qu.4.3.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0.025</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>T</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$tvalue</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0.05</mn></mrow></mstyle></math>@
qu.4.3.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><mn>0.05</mn><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>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'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tvalue</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0.10</mn></mrow></mstyle></math>@
qu.4.3.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>T</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tvalue</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0.100</mn></mrow></mstyle></math>@
qu.4.3.part.1.mode=Non Permuting Multiple Choice@
qu.4.3.part.1.display=vertical@
qu.4.3.part.1.answer=2@
qu.4.3.question=<p>If&nbsp;the random variable <em>T</em> follows a t distribution with $df degrees of freedom, which of the following statements is true?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.4.4.mode=Multiple Selection@
qu.4.4.name=Definitions 2: t Distribution@
qu.4.4.comment=@
qu.4.4.editing=useHTML@
qu.4.4.solution=@
qu.4.4.algorithm=@
qu.4.4.uid=e2dd3189-85d4-4cd1-8b4b-558cd96b9a92@
qu.4.4.info=  Course=Introductory Statistics;
  Topic=t Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.4.4.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.4.4.answer=1, 2@
qu.4.4.choice.1=The t distribution is used when <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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> is not known in the statistic <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><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></mrow></mstyle></math>, and must be estimated by s.@
qu.4.4.choice.2=The statistic <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&mu;</mi></mrow><mrow><mfrac><mi>s</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math> does not have a normal distribution.@
qu.4.4.choice.3=As the degrees of freedom increases, the shape of the t distribution becomes flatter.@
qu.4.4.choice.4=Replacing <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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> with s results in less uncertainty and variability in the statistic <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><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></mrow></mstyle></math>.@
qu.4.4.choice.5=If the random variable T follows a t distribution, the P(T < t) increases as the sample size increases, for any value of t.@
qu.4.4.fixed=@

qu.4.5.mode=Inline@
qu.4.5.name=Estimate range of P(T < t)@
qu.4.5.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>T</mi><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'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tvalue</mi></mrow></mfenced></mrow></mstyle></math>, we need to determine the area under a <em>t</em> distribution, with $df degrees of freedom, to the left of $tvalue.&nbsp; Using computer software, or approximating with a <em>t</em> distribution table, we can find this area to be $Prob.&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>T</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tvalue</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;is approximately 1.&nbsp; Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.4.5.editing=useHTML@
qu.4.5.solution=@
qu.4.5.algorithm=$df=range(6,10);
$tvalue=rand(4.0, 4.8, 4);
$Prob=studentst($df, $tvalue);
$ProbDisplay=decimal(4, $Prob);
$p=plotmaple("
f := Statistics[PDF](StudentT($df),x): 
p1 := plot(f, x=-3..$tvalue, colour=blue, filled=true): 
p2 := plot(f, x=$tvalue..3, colour=blue): 
p3 := plots[textplot]([$tvalue, -0.05, `$tvalue`], color=blue):
p4 := plots[textplot]([0.1, 0.12, `$ProbDisplay`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");@
qu.4.5.uid=141035a6-f2a7-426f-b39e-5ee066ea7a5a@
qu.4.5.info=  Course=Introductory Statistics;
  Topic=t Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.4.5.weighting=1@
qu.4.5.numbering=alpha@
qu.4.5.part.1.name=sro_id_1@
qu.4.5.part.1.editing=useHTML@
qu.4.5.part.1.choice.5=0.5@
qu.4.5.part.1.fixed=@
qu.4.5.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow></mrow></mstyle></math>@
qu.4.5.part.1.question=null@
qu.4.5.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>&infin;</mi></mrow></mstyle></math>@
qu.4.5.part.1.choice.2=0@
qu.4.5.part.1.choice.1=1@
qu.4.5.part.1.mode=Multiple Choice@
qu.4.5.part.1.display=vertical@
qu.4.5.part.1.answer=1@
qu.4.5.question=<p>If&nbsp;the random variable <em>T</em> follows a t distribution with $df degrees of freedom, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>T</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$tvalue</mi></mrow></mfenced></mrow></mstyle></math>is closest to which one of the following options?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.5.topic=F Distribution@

qu.5.1.mode=Inline@
qu.5.1.name=Estimate range of P(F > f)@
qu.5.1.comment=<p>To find 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><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>F</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><mn>4.0</mn></mrow></mfenced></mrow></mstyle></math>, for a random variable <em>F</em> that follows an <em>F</em> distribution with 7 and $DF2 degrees of freedom, we need to find the area under the curve to the right&nbsp;of 4.0.&nbsp; Using computer software, or approximating with an <em>F</em> distribution table, we can find this area to be $Prob.</p>
<p>Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$DF2=switch(rint(3), 7,8,9);
$Tail=maple("
X:=Statistics[CDF](FRatio(7,$DF2), 4.0):
X
");
$Prob=1-$Tail;
$ProbDisp=decimal(4, $Prob);
condition:gt($Prob,0.025);
condition:lt($Prob,0.05);
$p=plotmaple("
f := Statistics[PDF](FRatio(7, $DF2),x): 
p1 := plot(f, x=0..4.0, colour=blue): 
p2 := plot(f, x=4.0..6, colour=blue, filled=true): 
p3 := plots[textplot]([4.6, 0.02, `$ProbDisp`], color=black):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");@
qu.5.1.uid=a6d45947-b669-45e9-9400-2ce162cc2360@
qu.5.1.info=  Course=Introductory Statistics;
  Topic=F Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.5.1.weighting=1@
qu.5.1.numbering=alpha@
qu.5.1.part.1.name=sro_id_1@
qu.5.1.part.1.editing=useHTML@
qu.5.1.part.1.choice.5=0.005 to 0.010@
qu.5.1.part.1.fixed=@
qu.5.1.part.1.choice.4=0.010 to 0.025@
qu.5.1.part.1.question=null@
qu.5.1.part.1.choice.3=0.025 to 0.050@
qu.5.1.part.1.choice.2=0.050 to 0.100@
qu.5.1.part.1.choice.1=Greater than 0.100@
qu.5.1.part.1.mode=Non Permuting Multiple Choice@
qu.5.1.part.1.display=vertical@
qu.5.1.part.1.answer=3@
qu.5.1.question=<p>If the random variable <em>F</em> follows an F distribution with degrees of freedom 7 and $DF2, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>4.0</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;falls within which one of the following ranges?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.5.2.mode=Multiple Selection@
qu.5.2.name=Definitions 2: F Distribution@
qu.5.2.comment=@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=@
qu.5.2.uid=f69df335-2ecd-4826-8196-8700862157d7@
qu.5.2.info=  Course=Introductory Statistics;
  Topic=F Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.5.2.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.5.2.answer=1, 2, 3@
qu.5.2.choice.1=Since the F distribution arises as the distribution of a ratio, it relies on two degrees of freedom: one for the numerator and one for the denominator.@
qu.5.2.choice.2=If a random variable, X, follows an F distribution, X is a continuous random variable.@
qu.5.2.choice.3=In an F distribution, the median is often less than the mean.@
qu.5.2.choice.4=If a random variable X follows an F distribution, X cannot be less than 1.@
qu.5.2.choice.5=If a random variable X follows an F distribution, X can never be exactly equal to 1 .@
qu.5.2.fixed=@

qu.5.3.mode=Inline@
qu.5.3.name=Definitions 1&2: Random selection of True/False@
qu.5.3.comment=@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$k1=rint(10);
$k2=rint(10);
$k3=rint(10);
$k4=rint(10);
$k5=rint(10);
$z=maple("S := $k1,$k2,$k3,$k4,$k5:
floor( nops({S})/nops([S]) )");
condition: $z;
$a=("'The shape of the F distribution will change as the degrees of freedom change.'");
$b=("'A random variable, X, that follows an F distribution must be greater than or equal to 0.'");
$c=("'Since the F distribution arises as the distribution of a ratio, it relies on two degrees of freedom: one for the numerator and one for the denominator.'");
$d=("'If a random variable, X, follows an F distribution, X is a continuous random variable.'");
$e=("'In an F distribution, the median is often less than the mean.'");
$f=("'An F distribution is the result of the ratio of two normally distributed random variables.'");
$g=("'The F distribution is typically left skewed, and the extent of the skewness is determined by the degrees of freedom.'");
$h=("'There are three parameters in an F distribution: the degrees of freedom for the numerator, degrees of freedom for the denominator, and the mean.'");
$i=("'If a random variable X follows an F distribution, X cannot be less than 1.'");
$j=("'If a random variable X follows an F distribution, X can never be exactly equal to 1.'");
$Answers=["'True'","'True'","'True'","'True'","'True'","'False'","'False'","'False'","'False'","'False'"];
$Distractors=["'False'","'False'","'False'","'False'","'False'","'True'","'True'","'True'","'True'","'True'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A1=switch($k1, $Answers);
$D1=switch($k1, $Distractors);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A2=switch($k2, $Answers);
$D2=switch($k2, $Distractors);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A3=switch($k3, $Answers);
$D3=switch($k3, $Distractors);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A4=switch($k4, $Answers);
$D4=switch($k4, $Distractors);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$g,$h,$i,$j);
$A5=switch($k5, $Answers);
$D5=switch($k5, $Distractors);@
qu.5.3.uid=f0a4c711-8173-48a3-adb7-f154f601e728@
qu.5.3.info=  Course=Introductory Statistics;
  Topic=F Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.5.3.weighting=1,1,1,1,1@
qu.5.3.numbering=alpha@
qu.5.3.part.1.grader=exact@
qu.5.3.part.1.name=sro_id_1@
qu.5.3.part.1.editing=useHTML@
qu.5.3.part.1.display.permute=true@
qu.5.3.part.1.question=(Unset)@
qu.5.3.part.1.answer.2=$D1@
qu.5.3.part.1.answer.1=$A1@
qu.5.3.part.1.mode=List@
qu.5.3.part.1.display=menu@
qu.5.3.part.1.credit.2=0.0@
qu.5.3.part.1.credit.1=1.0@
qu.5.3.part.2.grader=exact@
qu.5.3.part.2.name=sro_id_2@
qu.5.3.part.2.editing=useHTML@
qu.5.3.part.2.display.permute=true@
qu.5.3.part.2.question=(Unset)@
qu.5.3.part.2.answer.2=$D2@
qu.5.3.part.2.answer.1=$A2@
qu.5.3.part.2.mode=List@
qu.5.3.part.2.display=menu@
qu.5.3.part.2.credit.2=0.0@
qu.5.3.part.2.credit.1=1.0@
qu.5.3.part.3.grader=exact@
qu.5.3.part.3.name=sro_id_3@
qu.5.3.part.3.editing=useHTML@
qu.5.3.part.3.display.permute=true@
qu.5.3.part.3.question=(Unset)@
qu.5.3.part.3.answer.2=$D3@
qu.5.3.part.3.answer.1=$A3@
qu.5.3.part.3.mode=List@
qu.5.3.part.3.display=menu@
qu.5.3.part.3.credit.2=0.0@
qu.5.3.part.3.credit.1=1.0@
qu.5.3.part.4.grader=exact@
qu.5.3.part.4.name=sro_id_4@
qu.5.3.part.4.editing=useHTML@
qu.5.3.part.4.display.permute=true@
qu.5.3.part.4.question=(Unset)@
qu.5.3.part.4.answer.2=$D4@
qu.5.3.part.4.answer.1=$A4@
qu.5.3.part.4.mode=List@
qu.5.3.part.4.display=menu@
qu.5.3.part.4.credit.2=0.0@
qu.5.3.part.4.credit.1=1.0@
qu.5.3.part.5.grader=exact@
qu.5.3.part.5.name=sro_id_5@
qu.5.3.part.5.editing=useHTML@
qu.5.3.part.5.display.permute=true@
qu.5.3.part.5.question=(Unset)@
qu.5.3.part.5.answer.2=$D5@
qu.5.3.part.5.answer.1=$A5@
qu.5.3.part.5.mode=List@
qu.5.3.part.5.display=menu@
qu.5.3.part.5.credit.2=0.0@
qu.5.3.part.5.credit.1=1.0@
qu.5.3.question=<p>Identify each of the following statements as either true or false.</p><p>&nbsp;</p><p>a)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> $Q1</p><p>&nbsp;</p><p>b)&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span> $Q2</p><p>&nbsp;</p><p>c)&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span> $Q3</p><p>&nbsp;</p><p>d)&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span> $Q4</p><p>&nbsp;</p><p>e)&nbsp;<span>&nbsp;</span><5><span>&nbsp;</span> $Q5</p>@

qu.5.4.mode=Inline@
qu.5.4.name=Estimate range of P(F < f)@
qu.5.4.comment=<p>To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>F</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><mn>2.0</mn></mrow></mfenced></mrow></mstyle></math>, we need to determine the area under the <em>F</em> distribution, with 17 and $DF2 degrees of freedom, to the left of 2.0.&nbsp; Using computer software, or approximating with an <em>F</em> distribution table, we can find this area to be $Prob.</p>
<p>Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.5.4.editing=useHTML@
qu.5.4.solution=@
qu.5.4.algorithm=$DF2=switch(rint(3), 2,3,4);
$Prob=maple("
X:=Statistics[CDF](FRatio(17,$DF2), 2.0):
X
");
$ProbDisp=decimal(4, $Prob);
$p=plotmaple("
f := Statistics[PDF](FRatio(17, $DF2),x): 
p1 := plot(f, x=0..2.0, colour=blue, filled=true): 
p2 := plot(f, x=2.0..4, colour=blue): 
p3 := plots[textplot]([1.0, 0.1, `$ProbDisp`], color=black):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");
condition:lt($Prob,0.90);@
qu.5.4.uid=c2141586-5530-4c32-a256-7406992f6592@
qu.5.4.info=  Course=Introductory Statistics;
  Topic=F Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.5.4.weighting=1@
qu.5.4.numbering=alpha@
qu.5.4.part.1.name=sro_id_1@
qu.5.4.part.1.editing=useHTML@
qu.5.4.part.1.fixed=@
qu.5.4.part.1.question=null@
qu.5.4.part.1.choice.3=Equal to 0.90@
qu.5.4.part.1.choice.2=Less than 0.90@
qu.5.4.part.1.choice.1=Greater than 0.90@
qu.5.4.part.1.mode=Multiple Choice@
qu.5.4.part.1.display=vertical@
qu.5.4.part.1.answer=2@
qu.5.4.question=<p>If the random variable <em>F</em> follows an F distribution with degrees of freedom 17 and $DF2, then&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>2.0</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;falls within which one of the following ranges?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.5.5.mode=Multiple Selection@
qu.5.5.name=Definitions 1: F Distribution@
qu.5.5.comment=@
qu.5.5.editing=useHTML@
qu.5.5.solution=@
qu.5.5.algorithm=@
qu.5.5.uid=567f4b46-c1c3-4b3b-ab57-416904917a2e@
qu.5.5.info=  Course=Introductory Statistics;
  Topic=F Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.5.5.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.5.5.answer=1, 2@
qu.5.5.choice.1=The shape of the F distribution will change as the degrees of freedom change.@
qu.5.5.choice.2=A random variable, X, that follows an F distribution must be greater than or equal to 0.@
qu.5.5.choice.3=An F distribution is the result of the ratio of two normally distributed random variables.@
qu.5.5.choice.4=The F distribution is typically left skewed, and the extent of the skewness is determined by the degrees of freedom.@
qu.5.5.choice.5=There are three parameters in an F distribution: the degrees of freedom for the numerator, degrees of freedom for the denominator, and the mean.@
qu.5.5.fixed=@

qu.6.topic=Chi-Square Distribution@

qu.6.1.mode=Inline@
qu.6.1.name=Estimate the range of the 90th percentile@
qu.6.1.comment=<p>To find the 90<sup>th</sup> percentile 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;distribution with 29 degrees of freedom, we need to find a value <em>X</em> on the distribution such that the area to the left of <em>X</em> is 0.90.&nbsp; Using computer software, or approximating with 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution table, we can find this value to be $PTL.</p>
<p>Graphically, this is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$PTL=maple("
X:=Statistics[Quantile](ChiSquare(29), 0.90):
X
");
$PTLDisp=decimal(4, $PTL);
$p=plotmaple("
f := Statistics[PDF](ChiSquare(29),x): 
p1 := plot(f, x=0..$PTL, colour=blue, filled=true): 
p2 := plot(f, x=$PTL..50, colour=blue): 
p3 := plots[textplot]([$PTL, -0.01, `$PTLDisp`], color=blue):
p4 := plots[textplot]([25.0, 0.01, `<-0.90->`], color=black):
plots[display]({p1,p2,p3,p4}), plotoptions='width=350,height=350'
");@
qu.6.1.uid=cfd67dcd-dd7f-4aba-8d6f-863394f09fed@
qu.6.1.info=  Course=Introductory Statistics;
  Topic=Chi-Square Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.6.1.weighting=1@
qu.6.1.numbering=alpha@
qu.6.1.part.1.name=sro_id_1@
qu.6.1.part.1.editing=useHTML@
qu.6.1.part.1.choice.5=29.05@
qu.6.1.part.1.fixed=@
qu.6.1.part.1.choice.4=14.26@
qu.6.1.part.1.question=null@
qu.6.1.part.1.choice.3=49.59@
qu.6.1.part.1.choice.2=39.09@
qu.6.1.part.1.choice.1=19.77@
qu.6.1.part.1.mode=Multiple Choice@
qu.6.1.part.1.display=vertical@
qu.6.1.part.1.answer=2@
qu.6.1.question=<p>The 90th percentile of&nbsp;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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution with 29 degrees of freedom is closest to&nbsp;which one of the following values?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span>&nbsp;</p>@

qu.6.2.mode=Multiple Selection@
qu.6.2.name=Definitions 1: Chi-Square Distribution@
qu.6.2.comment=@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=@
qu.6.2.uid=aa5ad429-19c0-4b72-a37d-fbe2f56273cd@
qu.6.2.info=  Course=Introductory Statistics;
  Topic=Chi-Square Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.6.2.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.6.2.answer=1, 2, 3@
qu.6.2.choice.1=The only parameter in 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> distribution is the degrees of freedom. @
qu.6.2.choice.2=As the degrees of freedom increases, the skewness of 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> distribution decreases. @
qu.6.2.choice.3=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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> distribution tends to be skewed to the right.@
qu.6.2.choice.4=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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> distribution with k degrees of freedom has a mean equal to k and variance equal 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><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>.@
qu.6.2.choice.5=If the random variable X follows 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> distribution, then X can take on any value between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='bold-italic' fontweight='bold' lspace='0.0em' rspace='0.0em'>to</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&infin;</mi></mrow></mrow></mstyle></math>.@
qu.6.2.fixed=@

qu.6.3.mode=Inline@
qu.6.3.name=Estimate range of (X^2 > x), 2 parts@
qu.6.3.comment=<p>a)&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>11.0</mn></mrow></mfenced></mrow></mstyle></math>, where <em>X</em> follows 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;distribution with $DFA degrees of freedom, we need to find the area under&nbsp;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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution to the right of 11.0.&nbsp; Using computer software, or approximating with 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution table, we can find this area to be $ProbA.</p>
<p>Graphically, this distribution looks like:</p>
<p>&nbsp;</p>
<p align="center">$P1</p>
<p>&nbsp;</p>
<p>b)&nbsp; To find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>11.0</mn></mrow></mfenced></mrow></mstyle></math>, when <em>X</em> now follows 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution with $DFB degrees of freedom, we need to determine the area under 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;distribution to the right of 11.0.&nbsp; Using computer software, or approximating with 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;distribution table, we can find this area to be $ProbB.</p>
<p>A plot of this distribution is as follows:</p>
<p>&nbsp;</p>
<p align="center">$P2</p>
<p>&nbsp;</p>
<p>Notice that the plots of 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;distribution for parts (a) and (b) are quite different.&nbsp; This is due to the change in shape of the distribution as the degrees of freedom change.</p>@
qu.6.3.editing=useHTML@
qu.6.3.solution=@
qu.6.3.algorithm=$DFA=range(26,29);
$DFB=switch(rint(2), 5,6);
$Tails=maple("
XA:=Statistics[CDF](ChiSquare($DFA), 11.0):
XB:=Statistics[CDF](ChiSquare($DFB), 11.0):
XA, XB
");
$TailA=switch(0, $Tails);
$TailB=switch(1, $Tails);
$ProbA=1-$TailA;
$ProbADisp=decimal(4, $ProbA);
$ProbB=1-$TailB;
$ProbBDisp=decimal(4, $ProbB);
condition:gt($ProbA,0.995);
condition:lt($ProbB,0.10);
condition:gt($ProbB,0.05);
$P1=plotmaple("
f := Statistics[PDF](ChiSquare($DFA),x): 
p1 := plot(f, x=0..11.0, colour=blue): 
p2 := plot(f, x=11.0..60.0, colour=blue, filled=true): 
p3 := plots[textplot]([25.0, 0.02, `$ProbADisp`], color=black):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");
$P2=plotmaple("
f := Statistics[PDF](ChiSquare($DFB),x): 
p1 := plot(f, x=0..11.0, colour=blue): 
p2 := plot(f, x=11.0..20.0, colour=blue, filled=true): 
p3 := plots[textplot]([13.0, 0.01, `$ProbBDisp`], color=black):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");@
qu.6.3.uid=ddc8e7a2-8feb-4a97-a522-364f4f888903@
qu.6.3.info=  Course=Introductory Statistics;
  Topic=Chi-Square Distribution;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.6.3.weighting=1,1@
qu.6.3.numbering=alpha@
qu.6.3.part.1.name=sro_id_1@
qu.6.3.part.1.editing=useHTML@
qu.6.3.part.1.choice.5=Less than 0.005@
qu.6.3.part.1.fixed=@
qu.6.3.part.1.choice.4=0.005 to 0.010@
qu.6.3.part.1.question=null@
qu.6.3.part.1.choice.3=0.975 to 0.990@
qu.6.3.part.1.choice.2=0.990 to 0.995@
qu.6.3.part.1.choice.1=Greater than 0.995@
qu.6.3.part.1.mode=Non Permuting Multiple Choice@
qu.6.3.part.1.display=vertical@
qu.6.3.part.1.answer=1@
qu.6.3.part.2.name=sro_id_2@
qu.6.3.part.2.editing=useHTML@
qu.6.3.part.2.choice.5=Less than 0.025@
qu.6.3.part.2.fixed=@
qu.6.3.part.2.choice.4=0.025 to 0.050@
qu.6.3.part.2.question=null@
qu.6.3.part.2.choice.3=0.050 to 0.100@
qu.6.3.part.2.choice.2=0.100 to 0.900@
qu.6.3.part.2.choice.1=0.950 to 0.900@
qu.6.3.part.2.mode=Non Permuting Multiple Choice@
qu.6.3.part.2.display=vertical@
qu.6.3.part.2.answer=3@
qu.6.3.question=<p>a)&nbsp; If the random variable <em>X</em> follows 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution with $DFA degrees of freedom, then&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>11.0</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;falls within which one of the following ranges?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span>b)&nbsp; If the random variable <em>X</em> follows 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution with $DFB degrees of freedom, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>X</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>11.0</mn></mrow></mfenced></mrow></mstyle></math>&nbsp;falls within which one of the following ranges?</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.6.4.mode=Multiple Selection@
qu.6.4.name=Definitions 2: Chi-Square Distribution@
qu.6.4.comment=@
qu.6.4.editing=useHTML@
qu.6.4.solution=@
qu.6.4.algorithm=@
qu.6.4.uid=9c8c678f-1776-44f3-b7c7-9ee9d67dbfe7@
qu.6.4.info=  Course=Introductory Statistics;
  Topic=Chi-Square Distribution;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.6.4.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>There may be more than one correct answer; select all that are true.</p>@
qu.6.4.answer=1, 2@
qu.6.4.choice.1=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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> random variable is the result of the sum of squared, independent standard normal random variables.@
qu.6.4.choice.2=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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> random variable, the variance is always greater than the mean.@
qu.6.4.choice.3=When there is 1 degree of freedom, the mean and variance of 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> distribution are the same.@
qu.6.4.choice.4=If 3 independent, standard normal random variables are square-rooted and then added together, the result is 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> random variable with 3 degrees of freedom.@
qu.6.4.choice.5=If the random variables <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>Z</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>and</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>Z</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math> are both normally distributed, with the same 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> and the same standard deviation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>, 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><msubsup><mi>Z</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msubsup><mi>Z</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mrow></mstyle></math>, with 2 degrees of freedom. @
qu.6.4.fixed=@

