qu.1.topic=Numerical Measures@

qu.1.1.mode=Inline@
qu.1.1.name=Calculate Standard Deviation for Random List with Outlier@
qu.1.1.comment=<p>a)&nbsp; To calculate the standard deviation of the data set, 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>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mover><mi>x</mi><mi>&macr;</mi></mover></mrow></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></msqrt></mrow></mrow></mstyle></math>, where the mean of the data set is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$TotalMean</mi></mrow></mstyle></math>.&nbsp; Substituting all 10 observations into the formula gives us <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$TotalMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$TotalMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$j</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$TotalMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>10</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$TotalSD</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>b)&nbsp; In this case, the outlying value is $j, as it is noticeably larger than the other values.&nbsp; Upon removing this observation, we get a new mean of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$RedMean</mi></mrow></mstyle></math>.&nbsp; Using this value in the formula for standard deviation seen in part (a), 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>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mfrac><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$RedMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$RedMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$i</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$RedMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>9</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ReducedSD</mi></mrow></mstyle></math></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=range(-10,10);
$b=range(-10,10);
$c=range(-10,10);
$d=range(-10,10);
$e=range(-10,10);
$f=range(-10,10);
$g=range(-10,10);
$h=range(-10,10);
$i=range(-10,10);
$j=range(80,90);
$TotalData=[$a, $b, $c, $d, $e, $f, $g, $h, $i, $j];
$ReducedData=[$a, $b, $c, $d, $e, $f, $g, $h, $i];
$Data=maple("
Total:=Statistics[StandardDeviation]($TotalData):
Reduced:=Statistics[StandardDeviation]($ReducedData):
TotalAvg:=Statistics[Mean]($TotalData):
RedAvg:=Statistics[Mean]($ReducedData):
Total, Reduced, TotalAvg, RedAvg
");
$TotalSD=switch(0, $Data);
$ReducedSD=switch(1, $Data);
$TotalMean=switch(2, $Data);
$RedMean=switch(3, $Data);@
qu.1.1.uid=71b70dd2-7c0d-40da-869d-73d98d8499a0@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Numerical Measures;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.1.weighting=1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.answer.units=@
qu.1.1.part.1.numStyle=   @
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.err=0.1@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=toler_abs@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$TotalSD@
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=0.1@
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=$ReducedSD@
qu.1.1.question=<p>Consider the following random sample of&nbsp;data:</p><p>&nbsp;</p><p>$a, $b, $c, $d, $e, $f, $g, $h, $i, $j</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the&nbsp;standard deviation&nbsp;of the sample data?</p><p>&nbsp;</p><p>Round&nbsp;your response to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; If the outlier is removed, what is the&nbsp;standard deviation&nbsp;of the remaining sample data?</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>Round&nbsp;your response to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Calculate Variance for Random List with Outlier@
qu.1.2.comment=<p>a)&nbsp; To calculate the variance of the data set, 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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mi mathcolor='#800080'>n</mi></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mover><mi>x</mi><mi>&macr;</mi></mover></mrow></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; For this data set, the mean is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$TotalMean</mi></mrow></mstyle></math>; substituting this into the equation for variance, along with the 10 observation values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$TotalMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$TotalMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$j</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$TotalMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>10</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$TotalVar</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>b)&nbsp; In this case, the outlier is $j, as this value is noticeably higher than the other values.&nbsp; Upon removing this value, the mean of the new data set is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$RedMean</mi></mrow></mstyle></math>.&nbsp; Using this value, the remaining 9 observations, and the formula for variance from part (a), 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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$RedMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$RedMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>$i</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$RedMean</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mn>9</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mstyle></math>= $ReducedVar</p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=range(-10,10);
$b=range(-10,10);
$c=range(-10,10);
$d=range(-10,10);
$e=range(-10,10);
$f=range(-10,10);
$g=range(-10,10);
$h=range(-10,10);
$i=range(-10,10);
$j=range(80,90);
$TotalData=[$a, $b, $c, $d, $e, $f, $g, $h, $i, $j];
$ReducedData=[$a, $b, $c, $d, $e, $f, $g, $h, $i];
$Data=maple("
Total:=Statistics[Variance]($TotalData):
Reduced:=Statistics[Variance]($ReducedData):
TotalAvg:=Statistics[Mean]($TotalData):
RedAvg:=Statistics[Mean]($ReducedData):
Total, Reduced, TotalAvg, RedAvg
");
$TotalVar=switch(0, $Data);
$ReducedVar=switch(1, $Data);
$TotalMean=switch(2, $Data);
$RedMean=switch(3, $Data);@
qu.1.2.uid=562e42aa-23a6-4792-8911-1199ce309231@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Numerical Measures;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.2.weighting=1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.answer.units=@
qu.1.2.part.1.numStyle=   @
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.showUnits=false@
qu.1.2.part.1.err=0.1@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.mode=Numeric@
qu.1.2.part.1.grading=toler_abs@
qu.1.2.part.1.negStyle=both@
qu.1.2.part.1.answer.num=$TotalVar@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.answer.units=@
qu.1.2.part.2.numStyle=   @
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.showUnits=false@
qu.1.2.part.2.err=0.1@
qu.1.2.part.2.question=(Unset)@
qu.1.2.part.2.mode=Numeric@
qu.1.2.part.2.grading=toler_abs@
qu.1.2.part.2.negStyle=both@
qu.1.2.part.2.answer.num=$ReducedVar@
qu.1.2.question=<p>Consider the following random sample of&nbsp;data:</p><p>&nbsp;</p><p>$a, $b, $c, $d, $e, $f, $g, $h, $i, $j</p><p>&nbsp;</p><p>a)&nbsp; What is the&nbsp;variance&nbsp;of the sample data?</p><p>&nbsp;</p><p>Round&nbsp;your response to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; If the outlier is removed, what is the&nbsp;variance&nbsp;of the remaining sample data?</span></p><p>&nbsp;</p><p><span>Round&nbsp;your response to&nbsp;at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.3.mode=Inline@
qu.1.3.name=Calculate Mean for Random List with Outlier@
qu.1.3.comment=<p>a)&nbsp; To calculate the mean, we use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>x</mi><msub><mi mathcolor='#0000ff'></mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Since there are 10 observations, this 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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>10</mn></mrow></munderover></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mn>10</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$j</mi></mrow></mfenced><mrow><mn>10</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$TotalMean</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>b)&nbsp; The outlier is the observation that is noticeably different from the other observations.&nbsp; In this case, the outlier is $j, as it is significantly larger than all the other observations.&nbsp; If this outlier is removed, and the mean of the remaining 9 observations is calculated, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>9</mn></mrow></munderover></mrow><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mrow><mrow><mn>9</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$a</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>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$i</mi></mrow></mfenced><mrow><mn>9</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ReducedMean</mi></mrow></mstyle></math></p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=range(-10,10);
$b=range(-10,10);
$c=range(-10,10);
$d=range(-10,10);
$e=range(-10,10);
$f=range(-10,10);
$g=range(-10,10);
$h=range(-10,10);
$i=range(-10,10);
$j=range(80,90);
$TotalData=[$a, $b, $c, $d, $e, $f, $g, $h, $i, $j];
$ReducedData=[$a, $b, $c, $d, $e, $f, $g, $h, $i];
$Data=maple("
Total:=Statistics[Mean]($TotalData):
Reduced:=Statistics[Mean]($ReducedData):
Total, Reduced
");
$TotalMean=switch(0, $Data);
$ReducedMean=switch(1, $Data);@
qu.1.3.uid=f90799a5-5378-495c-b04f-c1c37e84d087@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=Numerical Measures;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
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qu.1.3.part.1.err=0.01@
qu.1.3.part.1.question=(Unset)@
qu.1.3.part.1.mode=Numeric@
qu.1.3.part.1.grading=toler_abs@
qu.1.3.part.1.negStyle=both@
qu.1.3.part.1.answer.num=$TotalMean@
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qu.1.3.part.2.numStyle=   @
qu.1.3.part.2.editing=useHTML@
qu.1.3.part.2.showUnits=false@
qu.1.3.part.2.err=0.01@
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qu.1.3.part.2.mode=Numeric@
qu.1.3.part.2.grading=toler_abs@
qu.1.3.part.2.negStyle=both@
qu.1.3.part.2.answer.num=$ReducedMean@
qu.1.3.question=<p>Consider the following random sample of&nbsp;data:</p><p>&nbsp;</p><p>$a, $b, $c, $d, $e, $f, $g, $h, $i, $j</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the mean of the sample data?</p><p>&nbsp;</p><p>Round&nbsp;your response to&nbsp;at least&nbsp;2 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; If the outlier is removed, what is the mean of the remaining sample data?</span></p><p>&nbsp;</p><p><span>Round&nbsp;your response to&nbsp;at least&nbsp;2 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=Calculate Median for Random List with Outlier@
qu.1.4.comment=<p>a)&nbsp; To find the median, we first need to list the numbers in order from smallest to largest:</p>
<p align="center">$a&nbsp; $b&nbsp; $c&nbsp; $d&nbsp; $e&nbsp; $f&nbsp; $g&nbsp; $h&nbsp; $i&nbsp; $j</p>
<p align="left">Since there is an even number of observations, the median value will be the mean of the middle two numbers.&nbsp; Here, the median will be the average of observations 5 and 6, giving us <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Median</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$e</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>$f</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$TotalMedian</mi></mrow></mstyle></math>.</p>
<p align="left">&nbsp;</p>
<p align="left">b)&nbsp; The outlier in this data set is $j, as it is significantly larger than the other observations.&nbsp; If this observation is removed, and the data set is ordered from smallest to largest, we get:</p>
<p align="center">$a&nbsp; $b&nbsp; $c&nbsp; $d&nbsp; $e&nbsp; $f&nbsp; $g&nbsp; $h&nbsp; $i</p>
<p align="left">As there are now only 9 observations, the median is simply the middle value, which in this case is $e.</p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=range(1,3);
$b=range(4,7);
$c=range(8,10);
$d=range(11,13);
$e=range(14,17);
$f=range(18,20);
$g=range(21,23);
$h=range(24,27);
$i=range(28,30);
$j=range(100,110);
$TotalData=[$a, $b, $c, $d, $e, $f, $g, $h, $i, $j];
$ReducedData=[$a, $b, $c, $d, $e, $f, $g, $h, $i];
$Data=maple("
Total:=Statistics[Median]($TotalData):
Reduced:=Statistics[Median]($ReducedData):
Total, Reduced
");
$TotalMedian=switch(0, $Data);
$ReducedMedian=switch(1, $Data);@
qu.1.4.uid=b3495d61-28b1-497f-989b-4e0f36d47260@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=Numerical Measures;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.4.weighting=1,1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.answer.units=@
qu.1.4.part.1.numStyle=   @
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.showUnits=false@
qu.1.4.part.1.err=0.01@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.mode=Numeric@
qu.1.4.part.1.grading=toler_abs@
qu.1.4.part.1.negStyle=both@
qu.1.4.part.1.answer.num=$TotalMedian@
qu.1.4.part.2.name=sro_id_2@
qu.1.4.part.2.answer.units=@
qu.1.4.part.2.numStyle=   @
qu.1.4.part.2.editing=useHTML@
qu.1.4.part.2.showUnits=false@
qu.1.4.part.2.question=(Unset)@
qu.1.4.part.2.mode=Numeric@
qu.1.4.part.2.grading=exact_value@
qu.1.4.part.2.negStyle=both@
qu.1.4.part.2.answer.num=$ReducedMedian@
qu.1.4.question=<p>Consider the following random sample of&nbsp;data:</p><p>&nbsp;</p><p>$d, $h, $i, $e, $g, $b, $c, $a, $j, $f</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the median of the sample data?</p><p>&nbsp;</p><p>Round&nbsp;your response to&nbsp;2 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; If the outlier is removed, what is the median of the remaining sample data?</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.2.topic=Linear Transformations@

qu.2.1.mode=Multiple Selection@
qu.2.1.name=Effect of Addition of a Constant on Summary Statistics@
qu.2.1.comment=<p>Recall that the addition of a constant does not affect measures of variability.&nbsp; Therefore, the range, IQR, standard deviation and variance, which are all measure of variability, will not be affected by the addition of a constant.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$Percentile=range(70,90);
$ScriptOptions=["'th'","'st'","'nd'","'rd'","'th'","'th'","'th'","'th'","'th'","'th'"];
$Remain=maple("irem($Percentile, 10)");
$Superscript=switch($Remain, $ScriptOptions);@
qu.2.1.uid=9a35885d-ed85-4d92-80c4-a7933b031049@
qu.2.1.info=  Course=Introductory Statistics;
  Topic=Linear Transformations;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Concept;
@
qu.2.1.question=<p>Which of the following&nbsp;numerical measures&nbsp;would be affected by the addition of a constant to each value in the data set?&nbsp;</p>
<p>&nbsp;</p>
<p>Select all that&nbsp;may be&nbsp;true.</p>@
qu.2.1.answer=1, 2, 3, 7, 8, 10@
qu.2.1.choice.1=Mean@
qu.2.1.choice.2=Median@
qu.2.1.choice.3=Mode@
qu.2.1.choice.4=Variance@
qu.2.1.choice.5=Standard Deviation@
qu.2.1.choice.6=Range@
qu.2.1.choice.7=First Quartile@
qu.2.1.choice.8=Third Quartile@
qu.2.1.choice.9=IQR@
qu.2.1.choice.10=$Percentile$Superscript Percentile@
qu.2.1.fixed=@

qu.2.2.mode=Inline@
qu.2.2.name=Temperature Transformation - Celsius to Farenheit@
qu.2.2.comment=<p>To solve this question, we will make use of the linear transformation 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><msup><mi>x</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</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>bx</mi></mrow></mstyle></math>.</p>
<p>First, we must find the average January temperature in Buffalo, in Celsius, before we can convert it to Fahrenheit.&nbsp; Because the daily&nbsp;temperature in Buffalo is, on average, 3&deg;C warmer than it is in Guelph, we can use the linear transformation equation to get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mover><mi>x</mi><mi>&macr;</mi></mover></mrow><mrow><mi>Buffalo</mi></mrow><mrow><mi></mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>b</mi><msub><mrow><mover><mi>x</mi><mi>&macr;</mi></mover></mrow><mrow><mi>Guelph</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>3</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$AvgGuelph</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$AvgBuffaloC</mi></mrow></mstyle></math>.</p>
<p>Now, to convert the average temperature in Buffalo from Celsius to Fahrenheit, we can use the given conversion equation, 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><msub><mrow><mover><mi>x</mi><mi>&macr;</mi></mover></mrow><mrow><mi>F</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>9</mn><mrow><mn>5</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mrow><mover><mi>x</mi><mi>&macr;</mi></mover></mrow><mrow><mi>C</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>32</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mn>9</mn><mrow><mn>5</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$AvgBuffaloC</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>32</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$AvgBuffaloF</mi></mrow></mstyle></math></p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$AvgGuelph=rand(-8, -7, 3);
$AvgBuffaloC=$AvgGuelph + 3;
$AvgBuffaloF=((9/5)*$AvgBuffaloC) + 32;@
qu.2.2.uid=d7c797c3-0486-4847-a589-fafd657a64b9@
qu.2.2.info=  Course=Introductory Statistics;
  Topic=Linear Transformations;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
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qu.2.2.part.1.grading=toler_abs@
qu.2.2.part.1.negStyle=both@
qu.2.2.part.1.answer.num=$AvgBuffaloF@
qu.2.2.question=<p>In Guelph, Ontario, the average temperature in January is approximately $AvgGuelph &deg;Celsius.&nbsp; Across the lake, in Buffalo, New York, the daily temperature in January is 3&deg;Celsius&nbsp;higher on average&nbsp;than it is in Guelph, and is recorded in &deg;Fahrenheit.</p><p>&nbsp;</p><p>If the conversion from &deg;Celsius (C)&nbsp;to &deg;Fahrenheit (F) is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>9</mn><mrow><mn>5</mn></mrow></mfrac><mi>C</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>32</mn></mrow></mstyle></math>, what is the average January temperature in Buffalo, in &deg;Fahrenheit?</p><p>&nbsp;</p><p>Round&nbsp;your response to&nbsp;at least 3 decimal places.</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.2.3.mode=Inline@
qu.2.3.name=Transformation - Pounds to Kilograms@
qu.2.3.comment=<p>a)&nbsp; To find the mean total weight of the goalies in kilograms, we can use the linear transformation 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><msup><mrow><mover><mi>x</mi><mi>&macr;</mi></mover></mrow><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</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>b</mi><mrow><mover><mi>x</mi><mi>&macr;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; Because the average weight&nbsp;is given in pounds, we first need to convert this to kilograms, and then add on the weight of the equipment (which is already given in kilograms).&nbsp; Using the conversion from pounds to kilograms, we get the linear transformation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mover><mi>x</mi><mi>&macr;</mi></mover></mrow><mrow><mi>kg</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$Equipment</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0.45359</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$xbar</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$AvgTotal</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Recall that the addition of a constant does not affect measures of variability.&nbsp; Therefore, to find the variance of the weight of the goalies, in kilograms, we only need to consider the multiplier of&nbsp;0.45359.&nbsp; Using the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><msup><mi>x</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup><mrow><msubsup><mi>s</mi><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mstyle></math>, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mi>kg</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mn>0.45359</mn><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>$SD</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$VarianceTotal</mi></mrow></mstyle></math></p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$Measures=[175,208,201,219,217,190];
$Equipment=rand(11,14,4);
$Summary=maple("
XB:=Statistics[Mean]($Measures):
StDv:=Statistics[StandardDeviation]($Measures):
XB, StDv
");
$xbar=switch(0, $Summary);
$SD=switch(1, $Summary);
$AvgTotal=(0.45359*$xbar) + $Equipment;
$VarianceTotal=(0.45359^2)*($SD^2);@
qu.2.3.uid=915158a6-7d31-4f9e-bce8-b65e7883aacc@
qu.2.3.info=  Course=Introductory Statistics;
  Topic=Linear Transformations;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.2.3.weighting=1,1@
qu.2.3.numbering=alpha@
qu.2.3.part.1.name=sro_id_1@
qu.2.3.part.1.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=0.0010@
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=$AvgTotal@
qu.2.3.part.2.name=sro_id_2@
qu.2.3.part.2.answer.units=@
qu.2.3.part.2.numStyle=   @
qu.2.3.part.2.editing=useHTML@
qu.2.3.part.2.showUnits=false@
qu.2.3.part.2.err=0.0010@
qu.2.3.part.2.question=(Unset)@
qu.2.3.part.2.mode=Numeric@
qu.2.3.part.2.grading=toler_abs@
qu.2.3.part.2.negStyle=both@
qu.2.3.part.2.answer.num=$VarianceTotal@
qu.2.3.question=<p>The following observations represent the weights, in pounds (lbs), of a sample&nbsp;6 NHL goalies:</p><p>&nbsp;</p><p><table style="width: 374px; height: 58px" border="1" cellspacing="1" cellpadding="1" width="374" align="center">    <tbody>        <tr>            <td>Goalie 1</td>            <td>Goalie 2</td>            <td>Goalie 3</td>            <td>Goalie 4</td>            <td>Goalie 5</td>            <td>Goalie 6</td>        </tr>        <tr>            <td>175</td>            <td>201</td>            <td>208</td>            <td>219</td>            <td>217</td>            <td>190</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>Some summary statistics are: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SD</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>Before going on the ice, each goalie must put on approximately $Equipment kilograms (kg) of equipment.</p><p>&nbsp;</p><p>If 1 lbs = 0.45359 kg, then:</p><p>&nbsp;</p><p>a)&nbsp; What is the mean total weight (body weight plus equipment), in kilograms,&nbsp;of&nbsp;the goalies&nbsp;by the time&nbsp;they reach the ice?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>b)&nbsp; What is the variance in total weight (in kilograms)&nbsp;of&nbsp;the goalies&nbsp;by the time&nbsp;they reach the ice?</p><p>&nbsp;</p><p>Round your answer to&nbsp;at least 3 decimal places.</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p>@

qu.2.4.mode=Multiple Choice@
qu.2.4.name=Effect of Addition, Multiplication on Variance@
qu.2.4.comment=<p>Recall that the addition of a constant does not affect measures of variability.&nbsp; Therefore, the variance is only affected by the multiplier, which in this case is $b.&nbsp; Using the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><msup><mi>x</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msubsup><mi>s</mi><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mstyle></math>, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><msup><mi>x</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>$b</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msubsup><mi>s</mi><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$bb</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msubsup><mi>s</mi><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Hence, the variance is multiplied by $bb.</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$a=range(5,9);
$aa=$a^2;
$b=range(5,9);
$bb=$b^2;
condition:not(eq($a,$b));@
qu.2.4.uid=ab6a010f-f597-4afc-bf1e-c9cccbc60ad0@
qu.2.4.info=  Course=Introductory Statistics;
  Topic=Linear Transformations;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Concept;
@
qu.2.4.question=<p>Consider a data set with variance <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&sigma;</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>What is the effect on the variance if each observation in the data set is multiplied by $b, then increased by $a?</p>@
qu.2.4.answer=5@
qu.2.4.choice.1=The variance is multiplied by $b and increased by $a.@
qu.2.4.choice.2=The variance is increased by $a.@
qu.2.4.choice.3=The variance is multiplied by $a.@
qu.2.4.choice.4=The variance is multiplied by $bb and increased by $a.@
qu.2.4.choice.5=The variance is multiplied by $bb.@
qu.2.4.choice.6=The variance is multiplied by $b.@
qu.2.4.choice.7=The variance is increased by $aa and multiplied by $b.@
qu.2.4.fixed=@

qu.2.5.mode=Inline@
qu.2.5.name=Purchase Transformation - USD to CAD@
qu.2.5.comment=<p>a)&nbsp; To find the median price in Canadian dollars, we can use the linear transformation 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><msub><mi>M</mi><mrow><msup><mi>x</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>a</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>b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msub><mi>M</mi><mrow><mi>x</mi></mrow></msub></mrow></mrow></mstyle></math>, where <em>M</em> is the median value.&nbsp; Using the given exchange rate of 1 USD = $ExchangeRate CAD, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>M</mi><mrow><mi>CAD</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ExchangeRate</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msub><mi>M</mi><mrow><mi>USD</mi></mrow></msub></mrow></mrow></mstyle></math>&nbsp;(note that since there is no constant being added on, <em>a</em> = 0).&nbsp; Therefore, the median price paid in Canadian dollars is $MedianCAD.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To determine if the variability is higher or lower in Canadian dollars, we can use the relation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>s</mi><mrow><msup><mi>x</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>s</mi><mrow><mi>x</mi></mrow></msub></mrow></mstyle></math>.&nbsp; Since the standard deviation in USD is $StandDevUSD, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mi>CAD</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mi>$ExchangeRate</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$StandDevUSD</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$StandDevCAD</mi></mrow></mstyle></math>.&nbsp; Because the standard deviation in CAD is lower than in USD, we can conclude that the variability is lower in Canadian dollars.</p>@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=$MedianUSD=rand(122, 126, 5);
$StandDevUSD=rand(11, 13, 4);
$ExchangeRate=rand(0.97, 0.99, 3);
$MedianCAD=$ExchangeRate*$MedianUSD;
$StandDevCAD=$ExchangeRate*$StandDevUSD;@
qu.2.5.uid=ad60dc71-6e06-4645-bcae-102a83f91dca@
qu.2.5.info=  Course=Introductory Statistics;
  Topic=Linear Transformations;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Application;
@
qu.2.5.weighting=1,1@
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qu.2.5.part.1.name=sro_id_1@
qu.2.5.part.1.answer.units=@
qu.2.5.part.1.numStyle=  dollars @
qu.2.5.part.1.editing=useHTML@
qu.2.5.part.1.showUnits=false@
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qu.2.5.part.1.grading=toler_abs@
qu.2.5.part.1.negStyle=both@
qu.2.5.part.1.answer.num=$MedianCAD@
qu.2.5.part.2.grader=exact@
qu.2.5.part.2.name=sro_id_2@
qu.2.5.part.2.editing=useHTML@
qu.2.5.part.2.display.permute=true@
qu.2.5.part.2.question=(Unset)@
qu.2.5.part.2.answer.2=Lower@
qu.2.5.part.2.answer.1=Higher@
qu.2.5.part.2.mode=List@
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qu.2.5.part.2.credit.2=1.0@
qu.2.5.part.2.credit.1=0.0@
qu.2.5.question=<p>In an attempt to save money, you order 5 of your textbooks from an American-based company, and pay a&nbsp;median price&nbsp;of&nbsp;\\$ $MedianUSD, with a standard deviation of&nbsp;\\$ $StandDevUSD (both numbers are in US dollars (USD)).</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; If the current exchange rate is 1 USD =&nbsp; $ExchangeRate CAD, what was the median price you paid in Canadian dollars?</p><p>&nbsp;</p><p>Round your answer to&nbsp;2 decimal places.&nbsp; Do&nbsp;NOT include a dollar sign (\\$) in your response.</p><p>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p>b)&nbsp; Is the variability higher or lower in Canadian dollars?</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p>@

