qu.1.topic=Inference for Population Correlation@

qu.1.1.mode=Inline@
qu.1.1.name=Calculate r from raw data@
qu.1.1.comment=<p>In order to calculate <em>r</em>, we need to calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub></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><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub></mrow></mstyle></math>, where&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><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><msubsup><mi>x</mi><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><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><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mrow><mi>n</mi></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><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><msubsup><mi>y</mi><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><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><msub><mi>y</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mrow><mi>n</mi></mrow></mfrac></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><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><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><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>y</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><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><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><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><msub><mi>y</mi><mrow><mi>i</mi></mrow></msub></mrow></mfenced></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mstyle></math>.</p>
<p>Obtaining some basic summary statistics, 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><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><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SumX</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>10</mn></mrow></munderover><msubsup><mi>x</mi><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SumXSq</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><msub><mi>y</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SumY</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>10</mn></mrow></munderover><msubsup><mi>y</mi><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SumYSq</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><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><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>y</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SumXY</mi></mrow></mstyle></math>.&nbsp; Once these are substituted back into the formulas for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub></mrow></mstyle></math>, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SumXSq</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mi>$SumX</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mrow><mn>10</mn></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Sxx</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SumYSq</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mi>$SumY</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mrow><mn>10</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Syy</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SumXY</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$SumX</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SumY</mi></mrow></mfenced><mrow><mn>10</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Sxy</mi></mrow></mstyle></math>.&nbsp; Finally, we can substitute these values into the equation for <em>r</em>, such that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub></mrow><mrow><msqrt><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$Sxy</mi><mrow><msqrt><mrow><mi>$Sxx</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Syy</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$x1=rand(2,5,3);
$x2=rand(2,5,3);
$x3=rand(2,5,3);
$x4=rand(2,5,3);
$x5=rand(2,5,3);
$x6=rand(2,5,3);
$x7=rand(2,5,3);
$x8=rand(2,5,3);
$x9=rand(2,5,3);
$x10=rand(2,5,3);
$y1=rand(5,8,3);
$y2=rand(5,8,3);
$y3=rand(5,8,3);
$y4=rand(5,8,3);
$y5=rand(5,8,3);
$y6=rand(5,8,3);
$y7=rand(5,8,3);
$y8=rand(5,8,3);
$y9=rand(5,8,3);
$y10=rand(5,8,3);
$x=[$x1,$x2,$x3,$x4,$x5,$x6,$x7,$x8,$x9,$x10];
$y=[$y1,$y2,$y3,$y4,$y5,$y6,$y7,$y8,$y9,$y10];
$xSq=[$x1^2,$x2^2,$x3^2,$x4^2,$x5^2,$x6^2,$x7^2,$x8^2,$x9^2,$x10^2];
$ySq=[$y1^2,$y2^2,$y3^2,$y4^2,$y5^2,$y6^2,$y7^2,$y8^2,$y9^2,$y10^2];
$xy=[$x1*$y1,$x2*$y2,$x3*$y3,$x4*$y4,$x5*$y5,$x6*$y6,$x7*$y7,$x8*$y8,$x9*$y9,$x10*$y10];
$Data=maple("
R1:=Statistics[Correlation]($x, $y):
XSum:=convert($x, `+`):
YSum:=convert($y, `+`):
XsqSum:=convert($xSq, `+`):
YsqSum:=convert($ySq, `+`):
XYSum:=convert($xy, `+`):
R1, XSum, YSum, XsqSum, YsqSum, XYSum
");
$r=switch(0, $Data);
$SumX=switch(1, $Data);
$SumY=switch(2, $Data);
$SumXSq=switch(3, $Data);
$SumYSq=switch(4, $Data);
$SumXY=switch(5, $Data);
$Sxx=$SumXSq-(($SumX)^2/10);
$Syy=$SumYSq-(($SumY)^2/10);
$Sxy=$SumXY-($SumX*$SumY/10);@
qu.1.1.uid=0be40965-73a3-4c46-9235-dced24d529dc@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Correlation;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.1.weighting=1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.answer.units=@
qu.1.1.part.1.numStyle=   @
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.err=0.01@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=toler_abs@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$r@
qu.1.1.question=<p>The following observations are obtained from a random sample of 10 individuals:</p><p>&nbsp;</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Individual</strong></p>            </td>            <td>            <p align="center"><strong>x</strong></p>            </td>            <td>            <p align="center"><strong>y</strong></p>            </td>        </tr>        <tr>            <td>            <p align="center">1</p>            </td>            <td>            <p align="center">$x1</p>            </td>            <td>            <p align="center">$y1&nbsp;</p>            </td>        </tr>        <tr>            <td>            <p align="center">2</p>            </td>            <td>            <p align="center">&nbsp;$x2</p>            </td>            <td>            <p align="center">&nbsp;$y2</p>            </td>        </tr>        <tr>            <td>            <p align="center">3</p>            </td>            <td>            <p align="center">&nbsp;$x3</p>            </td>            <td>            <p align="center">&nbsp;$y3</p>            </td>        </tr>        <tr>            <td>            <p align="center">4</p>            </td>            <td>            <p align="center">&nbsp;$x4</p>            </td>            <td>            <p align="center">&nbsp;$y4</p>            </td>        </tr>        <tr>            <td>            <p align="center">5</p>            </td>            <td>            <p align="center">&nbsp;$x5</p>            </td>            <td>            <p align="center">&nbsp;$y5</p>            </td>        </tr>        <tr>            <td>            <p align="center">6</p>            </td>            <td>            <p align="center">&nbsp;$x6</p>            </td>            <td>            <p align="center">&nbsp;$y6</p>            </td>        </tr>        <tr>            <td>            <p align="center">7</p>            </td>            <td>            <p align="center">&nbsp;$x7</p>            </td>            <td>            <p align="center">&nbsp;$y7</p>            </td>        </tr>        <tr>            <td>            <p align="center">8</p>            </td>            <td>            <p align="center">&nbsp;$x8</p>            </td>            <td>            <p align="center">&nbsp;$y8</p>            </td>        </tr>        <tr>            <td>            <p align="center">9</p>            </td>            <td>            <p align="center">&nbsp;$x9</p>            </td>            <td>            <p align="center">&nbsp;$y9</p>            </td>        </tr>        <tr>            <td>            <p align="center">10</p>            </td>            <td>            <p align="center">&nbsp;$x10</p>            </td>            <td>            <p align="center">$y10&nbsp;</p>            </td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;Calculate the correlation coefficient, <em>r.</em></p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.2.mode=Multiple Selection@
qu.1.2.name=Definitions 1: Population Correlation@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
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qu.1.2.algorithm=@
qu.1.2.uid=63887516-8f61-4593-8bbc-17b0c5b658fb@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Inference for Population Correlation;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.2.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.1.2.answer=1, 2, 3@
qu.1.2.choice.1=There are no units associated with the correlation coefficient.@
qu.1.2.choice.2=Even if two variables are highly correlated, a change in one variable may not cause a change in the other.@
qu.1.2.choice.3=The test statistic for hypothesis testing on correlation is the same as that for hypothesis testing on the slope in linear regression.@
qu.1.2.choice.4=A large positive correlation implies a strong relationship between two variables, whereas a large negative correlation implies a weak relationship between two variables.@
qu.1.2.choice.5=Taking the square root of the correlation coefficient will give you the coefficient of determination.@
qu.1.2.fixed=@

qu.1.3.mode=Inline@
qu.1.3.name=Determine hypotheses, test statistic, conclusion for two-sided hypothesis test@
qu.1.3.comment=<p>a)&nbsp; The null and alternative hypotheses are given as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&rho;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&rho;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>, since we are specifying a two-sided alternative hypothesis (i.e. we are only testing if the population correlation is different from 0).</p>
<p>&nbsp;</p>
<p>b)&nbsp; The formula for the test statistic is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>r</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></msqrt></mrow></mfenced><mrow><msqrt><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></msqrt></mrow></mfenced><mrow><msqrt><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>$r</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp;&nbsp;Since we are conducting a two-sided test, the p-value is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tTest</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, where <em>t</em> follows a <em>t</em> distribution with $df degrees of freedom.&nbsp; Using computer software, we can find the area in one tail&nbsp;to be <em>$tail</em>, and therefore the p-value is <em>2 X $tail = $pvalue.</em>&nbsp; Since the p-value is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population correlation is different from 0.</p>@
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qu.1.3.algorithm=$n=range(20,30);
$df=$n-2;
$r=rand(-0.4, -0.1, 3);
$tTest=($r*sqrt($n-2))/sqrt(1-$r^2);
$tail=studentst($df, $tTest);
$pvalue=$tail*2;
condition:gt($pvalue,0.10);@
qu.1.3.uid=660c9b4a-0a44-4119-a58f-55102b22a727@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=Correlation, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.3.part.1.fixed=@
qu.1.3.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><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>r</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>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>r</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>0</mn></mrow></mstyle></math>@
qu.1.3.part.1.question=null@
qu.1.3.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><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>@
qu.1.3.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><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>r</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>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>r</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>@
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qu.1.3.part.2.err=0.0010@
qu.1.3.part.2.question=(Unset)@
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qu.1.3.part.2.negStyle=both@
qu.1.3.part.2.answer.num=$tTest@
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qu.1.3.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore conclude that there is no significant evidence the population correlation is different from 0.@
qu.1.3.part.3.choice.1=There is sufficient evidence to reject the null hypothesis, and therefore conclude that there is evidence the population correlation is not equal to 0.@
qu.1.3.part.3.mode=Multiple Choice@
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qu.1.3.part.3.answer=2@
qu.1.3.question=<p>A random sample of size $n is taken from a population, and for each individual in the sample measurements on two variables (<em>X</em>&nbsp;and <em>Y</em>) are obtained.&nbsp; The sample correlation of&nbsp;<em>X</em>&nbsp;and&nbsp;<em>Y</em> is calculated to be <em>r = $r</em>.</p><p>Test the null hypothesis that the population correlation is equal to 0, against the alternative hypothesis that it is not equal to 0.</p><p>&nbsp;</p><p>a)&nbsp; What are the appropriate null and alternative hypotheses?</p><p>&nbsp;</p><p><span>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Calculate the value of the test statistic.</span></p><p>&nbsp;</p><p><span>Round your response to at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp;&nbsp;What is&nbsp;the appropriate conclusion that can be made, at the 5% level of significance?</span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span>&nbsp;</span></span></span></p>@

qu.1.4.mode=Multiple Selection@
qu.1.4.name=Definitions 2: Population Correlation@
qu.1.4.comment=@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=@
qu.1.4.uid=77beea00-9917-4c49-82cf-cc26355988d6@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=Inference for Population Correlation;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.4.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.1.4.answer=1, 2@
qu.1.4.choice.1=The coefficient of determination is a proportion.@
qu.1.4.choice.2=The coefficient of determination is the proportion of the total variation of Y that can be explained by Y's linear relationship with X.@
qu.1.4.choice.3=The correlation coefficient measures the strength of any relationship between two variables.@
qu.1.4.choice.4=The correlation between two variables can never be exactly equal to 0.@
qu.1.4.choice.5=The correlation coefficient and the coefficient of determination are appropriate to use even when the relationship between two variables is not linear.@
qu.1.4.fixed=@

qu.1.5.mode=Inline@
qu.1.5.name=Calculate r^2 from summary statistics@
qu.1.5.comment=<p>To calculate the coefficient of determination, <em>r<sup>2</sup></em>,&nbsp;we can start&nbsp;by&nbsp;calculating&nbsp;the correlcation&nbsp;coefficient.<em>&nbsp; </em>The formual for <em>r</em>, the sample correlation, 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>r</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><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub></mrow><mrow><msqrt><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub></mrow></msqrt></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$Sxy</mi><mrow><msqrt><mrow><mi>$Sxx</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Syy</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$r</mi></mrow></mstyle></math>.&nbsp; Therefore, the coefficient of determination 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><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>$r</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$r2</mi></mrow></mstyle></math>.</p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$Sxy=rand(200, 250, 5);
$Sxx=rand(80, 99, 5);
$Syy=rand(2000, 2100, 5);
$r=$Sxy/sqrt($Sxx*$Syy);
$r2=$r^2;
condition:lt($r2,1.0);@
qu.1.5.uid=920561a7-d9b5-4fe7-9b26-2f06feca93d9@
qu.1.5.info=  Course=Introductory Statistics;
  Topic=Correlation;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.5.weighting=1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.answer.units=@
qu.1.5.part.1.numStyle=   @
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.showUnits=false@
qu.1.5.part.1.err=0.0010@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.mode=Numeric@
qu.1.5.part.1.grading=toler_abs@
qu.1.5.part.1.negStyle=both@
qu.1.5.part.1.answer.num=$r2@
qu.1.5.question=<p>The following summary statistics were obtained when measurements on&nbsp;two variables, <em>x</em> and <em>y, </em>were taken from&nbsp;a&nbsp;certain number of&nbsp;randomly selected individuals:</p><p>&nbsp;</p><p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Sxx</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Syy</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Sxy</mi></mrow></mstyle></math></p><p align="left">&nbsp;</p><p align="left">Calculate the coefficient of determination, <em>r<sup>2</sup></em>.</p><p align="left">&nbsp;</p><p align="left">Round your response to at least 3 decimal places.</p><p align="left"><span>&nbsp;</span><1><span>&nbsp;</span></p><p align="left"><span>&nbsp; </span></p>@

qu.1.6.mode=Inline@
qu.1.6.name=Determine degrees of freedom, test statistic, p-value for one-sided hypothesis test@
qu.1.6.comment=<p>a)&nbsp; If the null hypothesis is true, the test statistic will follow a <em>t</em> distribution with <em>n - 2</em> degrees of freedom.&nbsp; Since our sample size is <em>$n</em>, the degrees of freedom are <em>$n - 2 = $df</em>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The formula for the test statistic is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>r</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></msqrt></mrow></mfenced><mrow><msqrt><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>r</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$r</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></msqrt></mrow></mfenced><mrow><msqrt><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>$r</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The alternative hypothesis indicates that we are conducting a one-sided, upper tailed test.&nbsp; Therefore, the p-value is the area under the <em>t</em> distribution, with $df degrees of freedom, to the right of the test statistic.&nbsp; Using computer software, we can find this area to be <em>p-value = $pvalue.</em></p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$n=range(10,15);
$df=$n-2;
$r=rand(0.7, 0.8, 3);
$tTest=($r*sqrt($n-2))/sqrt(1-$r^2);
$pvalue=1-studentst($df, $tTest);
condition:lt($pvalue,0.025);@
qu.1.6.uid=89fa71fc-7c3c-415f-a1b6-cc2decba4b6e@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=Correlation, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.6.weighting=1,1,1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.answer.units=@
qu.1.6.part.1.numStyle=   @
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.showUnits=false@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.mode=Numeric@
qu.1.6.part.1.grading=exact_value@
qu.1.6.part.1.negStyle=both@
qu.1.6.part.1.answer.num=$df@
qu.1.6.part.2.name=sro_id_2@
qu.1.6.part.2.answer.units=@
qu.1.6.part.2.numStyle=   @
qu.1.6.part.2.editing=useHTML@
qu.1.6.part.2.showUnits=false@
qu.1.6.part.2.err=0.0010@
qu.1.6.part.2.question=(Unset)@
qu.1.6.part.2.mode=Numeric@
qu.1.6.part.2.grading=toler_abs@
qu.1.6.part.2.negStyle=both@
qu.1.6.part.2.answer.num=$tTest@
qu.1.6.part.3.name=sro_id_3@
qu.1.6.part.3.editing=useHTML@
qu.1.6.part.3.choice.5=p-value < 0.025@
qu.1.6.part.3.fixed=@
qu.1.6.part.3.choice.4=0.025 < p-value < 0.05@
qu.1.6.part.3.question=null@
qu.1.6.part.3.choice.3=0.05 < p-value < 0.10@
qu.1.6.part.3.choice.2=0.10 < p-value < 0.50@
qu.1.6.part.3.choice.1=p-value > 0.50@
qu.1.6.part.3.mode=Non Permuting Multiple Choice@
qu.1.6.part.3.display=vertical@
qu.1.6.part.3.answer=5@
qu.1.6.question=<p>A random sample of size $n is taken from a population, and for each individual in the sample measurements on two variables (<em>X</em>&nbsp; and <em>Y</em>) are obtained.&nbsp; The sample correlation of&nbsp;<em>X</em>&nbsp;and&nbsp;<em>Y</em> is calculated to be <em>r = $r</em>.</p><p>Carry out a hypothesis test on <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&rho;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0</mn></mrow></mstyle></math>against <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&rho;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn mathvariant='italic'>0</mn></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; If the null hypothesis is true, then the test statistic will follow a <em>t</em> distribution with what degrees of freedom?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Calculate the value of the test statistic.</span></p><p>&nbsp;</p><p><span>Round your response to at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; The p-value falls within which one of the following ranges:</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Calculate r from summary statistics@
qu.1.7.comment=<p>The formual for <em>r</em>, the sample correlation coefficient, 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>r</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><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub></mrow><mrow><msqrt><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub></mrow></msqrt></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>r</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$SSxy</mi></mrow><mrow><msqrt><mrow><mi>$SSxx</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SSyy</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$r</mi></mrow></mstyle></math>.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$SSxy=rand(310, 330, 4);
$SSxx=rand(50, 70, 3);
$SSyy=rand(2000, 2100, 5);
$r=$SSxy/sqrt($SSxx*$SSyy);
condition:lt($r,1.0);
condition:gt($r,-1.0);@
qu.1.7.uid=0c3084df-1fcb-4876-8c8f-2e7ddd258e30@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=Correlation;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.7.weighting=1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.answer.units=@
qu.1.7.part.1.numStyle=   @
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.showUnits=false@
qu.1.7.part.1.err=0.0010@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.mode=Numeric@
qu.1.7.part.1.grading=toler_abs@
qu.1.7.part.1.negStyle=both@
qu.1.7.part.1.answer.num=$r@
qu.1.7.question=<p>The following summary statistics were obtained when measurements on&nbsp;two variables,&nbsp;<em>X</em> and <em>Y, </em>were taken from 15 randomly selected individuals:</p><p>&nbsp;</p><p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSxx</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSyy</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSxy</mi></mrow></mstyle></math></p><p align="left">&nbsp;</p><p align="left">Calculate the correlation coefficient, <em>r</em>.</p><p align="left">&nbsp;</p><p align="left">Round your response to at least 3 decimal places.</p><p align="left"><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.8.mode=Inline@
qu.1.8.name=Definitions 1&2: Random selection of True/False@
qu.1.8.comment=@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.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=("'There are no units associated with the correlation coefficient.'");
$b=("'Even if two variables are highly correlated, a change in one variable may not cause a change in the other.'");
$c=("'The test statistic for hypothesis testing on correlation is the same as that for hypothesis testing on the slope in linear regression.'");
$d=("'The coefficient of determination is a proportion.'");
$e=("'The coefficient of determination is the proportion of total variation of Y that can be explained by its linear relationship with X.'");
$f=("'A large positive correlation implies a strong relationship between two variables, whereas a large negative correlation implies a weak relationship between two variables.'");
$g=("'Taking the square root of the correlation coefficient will give you the coefficient of determination.'");
$h=("'The correlation coefficient measures the strength of any relationship between two variables.'");
$i=("'The correlation between two variables can never be exactly equal to 0.'");
$j=("'The correlation coefficient and the coefficient of determination are appropriate to use even when the relationship between two variables is not linear.'");
$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, $Distractors);@
qu.1.8.uid=1e59653a-d9fc-4b4b-bae4-3e1946977ee5@
qu.1.8.info=  Course=Introductory Statistics;
  Topic=Inference for Population Correlation;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.8.weighting=1,1,1,1,1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.grader=exact@
qu.1.8.part.1.name=sro_id_1@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.display.permute=true@
qu.1.8.part.1.question=(Unset)@
qu.1.8.part.1.answer.2=$D1@
qu.1.8.part.1.answer.1=$A1@
qu.1.8.part.1.mode=List@
qu.1.8.part.1.display=menu@
qu.1.8.part.1.credit.2=0.0@
qu.1.8.part.1.credit.1=1.0@
qu.1.8.part.2.grader=exact@
qu.1.8.part.2.name=sro_id_2@
qu.1.8.part.2.editing=useHTML@
qu.1.8.part.2.display.permute=true@
qu.1.8.part.2.question=(Unset)@
qu.1.8.part.2.answer.2=$D2@
qu.1.8.part.2.answer.1=$A2@
qu.1.8.part.2.mode=List@
qu.1.8.part.2.display=menu@
qu.1.8.part.2.credit.2=0.0@
qu.1.8.part.2.credit.1=1.0@
qu.1.8.part.3.grader=exact@
qu.1.8.part.3.name=sro_id_3@
qu.1.8.part.3.editing=useHTML@
qu.1.8.part.3.display.permute=true@
qu.1.8.part.3.question=(Unset)@
qu.1.8.part.3.answer.2=$D3@
qu.1.8.part.3.answer.1=$A3@
qu.1.8.part.3.mode=List@
qu.1.8.part.3.display=menu@
qu.1.8.part.3.credit.2=0.0@
qu.1.8.part.3.credit.1=1.0@
qu.1.8.part.4.grader=exact@
qu.1.8.part.4.name=sro_id_4@
qu.1.8.part.4.editing=useHTML@
qu.1.8.part.4.display.permute=true@
qu.1.8.part.4.question=(Unset)@
qu.1.8.part.4.answer.2=$D4@
qu.1.8.part.4.answer.1=$A4@
qu.1.8.part.4.mode=List@
qu.1.8.part.4.display=menu@
qu.1.8.part.4.credit.2=0.0@
qu.1.8.part.4.credit.1=1.0@
qu.1.8.part.5.grader=exact@
qu.1.8.part.5.name=sro_id_5@
qu.1.8.part.5.editing=useHTML@
qu.1.8.part.5.display.permute=true@
qu.1.8.part.5.question=(Unset)@
qu.1.8.part.5.answer.2=$D5@
qu.1.8.part.5.answer.1=$A5@
qu.1.8.part.5.mode=List@
qu.1.8.part.5.display=menu@
qu.1.8.part.5.credit.2=0.0@
qu.1.8.part.5.credit.1=1.0@
qu.1.8.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>@

