qu.1.topic=Simple Linear Regression@

qu.1.1.mode=Inline@
qu.1.1.name=Calculate predicted value of Y & residual value@
qu.1.1.comment=<p>a)&nbsp; The fitted regression line is given as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi></mrow></mstyle></math>, and therefore to find the&nbsp;predicted value of <em>Y</em> at the third observation, when <em>X = $x3</em>, we simply substitute this value into the equation.&nbsp; This results in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$x3</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$yfitted</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The residual for the third observation 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><msub><mi>e</mi><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>Y</mi><mrow><mn>3</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>3</mn></mrow></msub></mrow></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>Y</mi><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math>is the observed value of <em>Y,</em>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math>&nbsp;is the predicted value of <em>Y</em>, found in part (a).&nbsp; Therefore, the residual for the third observation (i.e. when <em>X = $x3</em>) is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>e</mi><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$y3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$yfitted</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$residual</mi></mrow></mstyle></math>.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$x1=rand(9,10,3);
$x2=rand(4,5,3);
$x3=rand(6,7,3);
$x4=rand(10,11,3);
$x5=rand(8,9,3);
$x6=rand(10,11,3);
$x7=rand(4,5,3);
$x8=rand(8,9,3);
$x9=rand(7,8,3);
$x10=rand(9,10,3);
$y1=rand(5,6,3);
$y2=rand(3,4,3);
$y3=rand(4,5,3);
$y4=rand(5,6,3);
$y5=rand(4,5,3);
$y6=rand(5,6,3);
$y7=rand(3,4,3);
$y8=rand(5,6,3);
$y9=rand(4,5,3);
$y10=rand(5,6,3);
$xList=[$x1,$x2,$x3,$x4,$x5,$x6,$x7,$x8,$x9,$x10];
$yList=[$y1,$y2,$y3,$y4,$y5,$y6,$y7,$y8,$y9,$y10];
$m=maple("
map(rhs, Statistics[Fit](a*x+b, $xList, $yList, x, output=parametervalues))
");
$a=decimal(2, switch(0,$m));
$b=decimal(2, switch(1,$m));
$yfitted=$a*$x3 + $b;
$residual=$y3 - $yfitted;@
qu.1.1.uid=f7788d19-7b43-40de-91a5-1f15fb5b6a6b@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Simple Linear Regression;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.1.weighting=1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.answer.units=@
qu.1.1.part.1.numStyle=   @
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.err=0.0010@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=toler_abs@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$yfitted@
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=0.0010@
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=$residual@
qu.1.1.question=<p>A random sample of 10 individuals is selected from a population, and measurements on two variables (<em>X</em> and <em>Y</em>) are obtained, as seen in the table below.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Individual</strong></p>            </td>            <td>            <p align="center"><em><strong>X</strong></em></p>            </td>            <td>            <p align="center"><em><strong>Y</strong></em></p>            </td>        </tr>        <tr>            <td>            <p align="center">1</p>            </td>            <td>            <p align="center">$x1</p>            </td>            <td>            <p align="center">$y1</p>            </td>        </tr>        <tr>            <td>            <p align="center">2</p>            </td>            <td>            <p align="center">$x2</p>            </td>            <td>            <p align="center">$y2</p>            </td>        </tr>        <tr>            <td>            <p align="center">3</p>            </td>            <td>            <p align="center">$x3</p>            </td>            <td>            <p align="center">$y3</p>            </td>        </tr>        <tr>            <td>            <p align="center">4</p>            </td>            <td>            <p align="center">$x4</p>            </td>            <td>            <p align="center">$y4</p>            </td>        </tr>        <tr>            <td>            <p align="center">5</p>            </td>            <td>            <p align="center">$x5</p>            </td>            <td>            <p align="center">$y5</p>            </td>        </tr>        <tr>            <td>            <p align="center">6</p>            </td>            <td>            <p align="center">$x6</p>            </td>            <td>            <p align="center">$y6</p>            </td>        </tr>        <tr>            <td>            <p align="center">7</p>            </td>            <td>            <p align="center">$x7</p>            </td>            <td>            <p align="center">$y7</p>            </td>        </tr>        <tr>            <td>            <p align="center">8</p>            </td>            <td>            <p align="center">$x8</p>            </td>            <td>            <p align="center">$y8</p>            </td>        </tr>        <tr>            <td>            <p align="center">9</p>            </td>            <td>            <p align="center">$x9</p>            </td>            <td>            <p align="center">$y9</p>            </td>        </tr>        <tr>            <td>            <p align="center">10</p>            </td>            <td>            <p align="center">$x10</p>            </td>            <td>            <p align="center">$y10</p>            </td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>The fitted regression line for the linear relationship between <em>X</em>&nbsp; and <em>Y</em> was found to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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><mi>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi></mrow></mstyle></math>.</p><p>Assuming all the model assumptions are met, and the inference procedures are valid, then:</p><p>&nbsp;</p><p>a)&nbsp; What is the predicted value of <em>Y</em> for the third individual?</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the value of the residual for the third individual?</span></p><p>&nbsp;</p><p><span>Round your response to at least 3 decimal places.</span></p><p>&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span></p>@

qu.1.2.mode=Multiple Selection@
qu.1.2.name=Definitions 1: Simple Linear Regression@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=@
qu.1.2.uid=137767f6-7151-4913-a21c-cfdbe3bae9ef@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Simple Linear Regression;
  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@
qu.1.2.choice.1=The purpose of linear regression is to investigate if there exists a linear relationship between a response variable and one or more explanatory variables.@
qu.1.2.choice.2=A 'residual' is the difference between an observed and predicted value of Y, for a particular value of X.@
qu.1.2.choice.3=If the purpose of our regression model is prediction, it does not matter which variables we define as the explanatory and response variable.@
qu.1.2.choice.4=The observed values of Y will fall on the estimated regression line, while the predicted values of Y will vary around the regression line.@
qu.1.2.choice.5=The sum of the residuals must be 1.@
qu.1.2.fixed=@

qu.1.3.mode=Inline@
qu.1.3.name=Calculate point estimate, standard error for confidence and prediction interval@
qu.1.3.comment=<p>a)&nbsp; The point estimate for the mean value of <em>Y</em> when <em>X</em> = $xvalue (i.e. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mover><mi>&mu;</mi><mi>&#x005e;</mi></mover></mrow><mrow><mi>Y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>X</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow></msub></mrow></mstyle></math>) is the same as the point estimate for the predicted value of <em>Y </em>(i.e. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>) when <em>X = $xvalue</em>.&nbsp; In either case, we find the estimate by simply substituting <em>X = $xvalue</em> into the regression equation, so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>&mu;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mi>Y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>X</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b0</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>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$xvalue</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$PE</mi></mrow></mstyle></math>.</p>
<p>b)&nbsp; The standard error for a 95% confidence interval for the mean of <em>Y</em> when <em>X = $xvalue</em>&nbsp; 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>SE</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><msub><mover><mrow><mi>&mu;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mi>Y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>X</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow></msub><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>s</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>X</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub></mrow></mfrac></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values results in a standard error of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><msub><mover><mrow><mi>&mu;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mi>Y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>X</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow></msub><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><mi>$s2</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xvalue</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mrow><mi>$SSxx</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SEmean</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The standard error for a 95% confidence interval for the predicted value of <em>Y</em>, for <em>X = $xvalue</em>, is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>Y</mi><mrow><mi>Pred</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>s</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>X</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub></mrow></mfrac></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get a value for the standard error for the predicted value of <em>Y</em>&nbsp; to be calculated as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>Y</mi><mrow><mi>Pred</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$s2</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mn>1</mn><mrow><mi>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mi>$xvalue</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mrow><mi>$SSxx</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SEpred</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$SSxy=rand(1000,1200,5);
$SSxx=rand(800,999,4);
$SSyy=rand(1500,1600,5);
$s2=rand(40,50,4);
$n=range(20,30);
$df=$n-2;
$xbar=rand(150,200,4);
$ybar=rand(450,500,4);
$b1=$SSxy/$SSxx;
$b0=$ybar-($b1*$xbar);
$xvalue=$xbar+rand(10,15,3);
$SEmean=sqrt($s2)*sqrt((1/$n)+($xvalue-$xbar)^2/$SSxx);
$SEpred=sqrt($s2)*sqrt(1+(1/$n)+($xvalue-$xbar)^2/$SSxx);
$PE=$b0 + $b1*$xvalue;@
qu.1.3.uid=2f8bfcb3-a8a9-4130-a9bb-c52eeaf13968@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=Simple Linear Regression;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.3.weighting=1,1,1,1@
qu.1.3.numbering=alpha@
qu.1.3.part.1.name=sro_id_1@
qu.1.3.part.1.answer.units=@
qu.1.3.part.1.numStyle=   @
qu.1.3.part.1.editing=useHTML@
qu.1.3.part.1.showUnits=false@
qu.1.3.part.1.err=0.0010@
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=$PE@
qu.1.3.part.2.grader=exact@
qu.1.3.part.2.name=sro_id_2@
qu.1.3.part.2.editing=useHTML@
qu.1.3.part.2.display.permute=true@
qu.1.3.part.2.question=(Unset)@
qu.1.3.part.2.answer.2=No, the point estimates are not exactly the same.@
qu.1.3.part.2.answer.1=Yes, the point estimates are exactly the same.@
qu.1.3.part.2.mode=List@
qu.1.3.part.2.display=menu@
qu.1.3.part.2.credit.2=0.0@
qu.1.3.part.2.credit.1=1.0@
qu.1.3.part.3.name=sro_id_3@
qu.1.3.part.3.answer.units=@
qu.1.3.part.3.numStyle=   @
qu.1.3.part.3.editing=useHTML@
qu.1.3.part.3.showUnits=false@
qu.1.3.part.3.err=0.01@
qu.1.3.part.3.question=(Unset)@
qu.1.3.part.3.mode=Numeric@
qu.1.3.part.3.grading=toler_abs@
qu.1.3.part.3.negStyle=both@
qu.1.3.part.3.answer.num=$SEmean@
qu.1.3.part.4.name=sro_id_4@
qu.1.3.part.4.answer.units=@
qu.1.3.part.4.numStyle=   @
qu.1.3.part.4.editing=useHTML@
qu.1.3.part.4.showUnits=false@
qu.1.3.part.4.err=0.01@
qu.1.3.part.4.question=(Unset)@
qu.1.3.part.4.mode=Numeric@
qu.1.3.part.4.grading=toler_abs@
qu.1.3.part.4.negStyle=both@
qu.1.3.part.4.answer.num=$SEpred@
qu.1.3.question=<p>The following summary statistics are obtained when a random sample of $n individuals are drawn from a population, and measurements on two variables (<em>X</em>&nbsp; and <em>Y</em>) are obtained on each individual:</p><p>&nbsp;</p><p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>Y</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ybar</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSxy</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSxx</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSyy</mi></mrow></mstyle></math></p><p>&nbsp;</p><p>The subsequent regression line for examining the linear relationship between <em>X</em> and <em>Y</em> is found to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b0</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>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>&nbsp;</p><p><span>&nbsp;a)&nbsp;i)&nbsp;&nbsp;Calculate a point estimate for t</span><span>he mean value&nbsp;of <em>Y </em>when<em> X&nbsp;</em>is equal to <em>$xvalue</em>.</span></p><p>&nbsp;</p><p><span>Round your response to at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p><span><span>ii)&nbsp; Is this estimate exactly the same as the point estimate of a single predicted value of <em>Y</em> when <em>X</em> is equal to <em>$xvalue</em>?</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>b)&nbsp; Calculate the standard error for the estimated&nbsp;mean value of <em>Y</em> when <em>X = $xvalue</em>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>SE</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><msub><mover><mrow><mi>&mu;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mi>Y</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><msup><mi>X</mi><mrow><mo lspace='0.1666667em' rspace='0.1666667em'>&ast;</mo></mrow></msup></mrow></msub><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</span></span></span></p><p>&nbsp;</p><p><span><span><span>Round your response to at least 3 decimal places.</span></span></span></p><p><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>c)&nbsp; Calculate the standard error of the predicted value of <em>Y</em> when <em>X = $xvalue, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mi>Y</mi><mrow><mi>Pred</mi></mrow></msub></mrow></mfenced></mrow></mstyle></math></em>.</span></span></span></p><p>&nbsp;</p><p><span><span><span>Round your response to at least 3 decimal places.</span></span></span></p><p><span><span><span><span>&nbsp;</span><4><span>&nbsp;</span></span></span></span></p>@

qu.1.4.mode=Multiple Selection@
qu.1.4.name=Definitions 2: Simple Linear Regression@
qu.1.4.comment=@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=@
qu.1.4.uid=34814c63-a38e-4965-834a-93bae6eadc37@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=Simple Linear Regression;
  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, 3@
qu.1.4.choice.1=The Method of Least Squares is a method of fitting a regression line to the data, 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><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><msubsup><mi>e</mi><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math> is minimized.@
qu.1.4.choice.2=If the coefficient of determination is very large, it would be reasonable to expect that a hypothesis test on the slope would result in a very small p-value.@
qu.1.4.choice.3=A transformation may be applied to the response variable or the explanatory variable (or both), in an attempt to establish a linear relationship between the variables.@
qu.1.4.choice.4=If a confidence interval for the true value of the slope contains 1, we can be certain that there is no evidence of a linear relationship between X and Y.@
qu.1.4.choice.5=If a plot of the residuals versus fitted values shows a distinct straight-line (i.e. linear) pattern, then the model assumptions have been verified and the inference procedures are valid.@
qu.1.4.fixed=@

qu.1.5.mode=Inline@
qu.1.5.name=Calculate slope, margin of error for 95% confidence interval@
qu.1.5.comment=<p>a)&nbsp; To estimate the slope, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&beta;</mi><mrow><mn>1</mn></mrow></msub></mrow></mstyle></math>,&nbsp;we use the equation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values gives us <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$SSxy</mi><mrow><mi>$SSxx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b1</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To estimate the margin of error for a 95% confidence interval for the true slope, we need to first esimate the standard error for our estimate of the slope, given as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>s</mi><mrow><msqrt><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Once the appropriate values have been substituted in, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><msqrt><mrow><mi>$s2</mi></mrow></msqrt><mrow><msqrt><mrow><mi>$SSxx</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SEb1</mi></mrow></mstyle></math>.&nbsp; This value is then used to estimate the margin of error, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>, where for a <em>t</em> distribution with <em>n - 2 = $df</em> degrees of freedom,&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tAlpha2</mi></mrow></mstyle></math>.&nbsp; Finally, we can calculate the margin of error as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SEb1</mi></mrow></mstyle></math>.</p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$n=range(15,20);
$df=$n-2;
$ybar=rand(35,40,4);
$xbar=rand(25,30,4);
$SSxy=rand(100,150,4);
$SSxx=rand(200,250,4);
$b1=$SSxy/$SSxx;
$b0=$ybar-($b1*$xbar);
$s2=rand(20,30,3);
$SEb1=sqrt($s2/$SSxx);
$tAlpha2=invstudentst($df, 0.975);
$MEb1=$tAlpha2*$SEb1;@
qu.1.5.uid=e2e43b57-31b7-4a8d-b28c-66646aea6b82@
qu.1.5.info=  Course=Introductory Statistics;
  Topic=Simple Linear Regression;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.5.weighting=1,1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.answer.units=@
qu.1.5.part.1.numStyle=   @
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.showUnits=false@
qu.1.5.part.1.err=0.0010@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.mode=Numeric@
qu.1.5.part.1.grading=toler_abs@
qu.1.5.part.1.negStyle=both@
qu.1.5.part.1.answer.num=$b1@
qu.1.5.part.2.name=sro_id_2@
qu.1.5.part.2.answer.units=@
qu.1.5.part.2.numStyle=   @
qu.1.5.part.2.editing=useHTML@
qu.1.5.part.2.showUnits=false@
qu.1.5.part.2.err=0.01@
qu.1.5.part.2.question=(Unset)@
qu.1.5.part.2.mode=Numeric@
qu.1.5.part.2.grading=toler_abs@
qu.1.5.part.2.negStyle=both@
qu.1.5.part.2.answer.num=$MEb1@
qu.1.5.question=<p>Measurements on two variables, <em>X</em> and <em>Y</em>, were taken from&nbsp;$n individuals, with the following summary statistics:</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><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>Y</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ybar</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$SSxy</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSxx</mi></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi></mrow></mstyle></math></p><p>&nbsp;</p><p>Use this information to&nbsp;make inferences on&nbsp;the&nbsp;slope for the least squares regression line, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&beta;</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>&beta;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&epsilon;</mi></mrow></mrow></mstyle></math>.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the point estimate of the true slope, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&beta;</mi><mrow><mn>1</mn></mrow></msub></mrow></mstyle></math>?</p><p>&nbsp;</p><p>Round your response to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the margin of error for a 95% confidence interval for the true slope?</span></p><p>&nbsp;</p><p><span>Round your response to at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.6.mode=Inline@
qu.1.6.name=Calculate predicted value of Y@
qu.1.6.comment=<p>a)&nbsp; The fitted regression line is given as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi></mrow></mstyle></math>, and therefore to find the&nbsp;predicted value of <em>Y</em> when <em>X = $xvalue</em>, we simply substitute this value into the equation.&nbsp; This results in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$xvalue</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$yfitted</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; From the plot of the data, it is clear that an <em>X</em> value of $xvalue is well outside the range of observed data.&nbsp; To predict a value of <em>Y</em>&nbsp; at <em>X = $xvalue</em> is an example of extrapolation, which should be avoided as it can lead to misleading results.&nbsp; Therefore, it is not reasonable to try and predict <em>Y</em> at <em>X = $xvalue.</em></p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$x1=rand(9,10,3);
$x2=rand(4,5,3);
$x3=rand(6,7,3);
$x4=rand(10,11,3);
$x5=rand(8,9,3);
$x6=rand(10,11,3);
$x7=rand(4,5,3);
$x8=rand(8,9,3);
$x9=rand(7,8,3);
$x10=rand(9,10,3);
$y1=rand(5,6,3);
$y2=rand(3,4,3);
$y3=rand(4,5,3);
$y4=rand(5,6,3);
$y5=rand(4,5,3);
$y6=rand(5,6,3);
$y7=rand(3,4,3);
$y8=rand(5,6,3);
$y9=rand(4,5,3);
$y10=rand(5,6,3);
$xList=[$x1,$x2,$x3,$x4,$x5,$x6,$x7,$x8,$x9,$x10];
$yList=[$y1,$y2,$y3,$y4,$y5,$y6,$y7,$y8,$y9,$y10];
$m=maple("
map(rhs, Statistics[Fit](a*x+b, $xList, $yList, x, output=parametervalues))
");
$a=decimal(2, switch(0,$m));
$b=decimal(2, switch(1,$m));
$xvalue=range(30,40);
$yfitted=$a*$xvalue + $b;
$myplot=plotmaple("
p1:=plot($a*x+$b, x=0.0..12):
p2:=plots[pointplot]($xList,$yList, style=point, symbol=circle):
plots[display]({p1,p2}), plotoptions='width=350,height=350':
");@
qu.1.6.uid=7857a01b-74ff-4f4b-a27b-1f409da2d29b@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=Simple Linear Regression;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=Scatter plot with regression line;
  Type=Calculation;
@
qu.1.6.weighting=1,1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.answer.units=@
qu.1.6.part.1.numStyle=   @
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.showUnits=false@
qu.1.6.part.1.err=0.01@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.mode=Numeric@
qu.1.6.part.1.grading=toler_abs@
qu.1.6.part.1.negStyle=both@
qu.1.6.part.1.answer.num=$yfitted@
qu.1.6.part.2.name=sro_id_2@
qu.1.6.part.2.editing=useHTML@
qu.1.6.part.2.fixed=@
qu.1.6.part.2.choice.4=Yes.&nbsp; This is an example of interpolation, and is a reasonable way to predict unobserved values when the linear regression model fits the data well.@
qu.1.6.part.2.question=null@
qu.1.6.part.2.choice.3=No.&nbsp; This is an example of interpolation, and should be avoided as the results can be inaccurate and unstable.@
qu.1.6.part.2.choice.2=Yes.&nbsp; This is an example of extrapolation, and is a valid way to extend the linear regression line to unobserved values.@
qu.1.6.part.2.choice.1=No.&nbsp; This is an example of extrapolation, and should be avoided as it can have misleading results.@
qu.1.6.part.2.mode=Multiple Choice@
qu.1.6.part.2.display=vertical@
qu.1.6.part.2.answer=1@
qu.1.6.question=<p>Researchers are interested in investigating whether or not there is a relationship between two variables, <em>X</em>&nbsp; and<em>&nbsp;Y</em>.&nbsp; To do so, they obtained a random sample of 10 individuals, measured values of <em>X</em> and <em>Y</em> on each individual, and fit a linear regression model to the resulting data, as seen in the plot below.</p><p>&nbsp;</p><p align="center">$myplot</p><p>&nbsp;</p><p>The fitted regression line was found to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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><mi>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi></mrow></mstyle></math>.</p><p>Assuming all the model assumptions are met, and the inference procedures are valid, then:</p><p>&nbsp;</p><p>a)&nbsp; What is the predicted value of <em>Y</em>, at <em>X = $xvalue</em>?</p><p>&nbsp;</p><p>Round your response to at least&nbsp;2 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Is it reasonable to use the regression line to predict <em>Y</em> at <em>X = $xvalue</em>?</span></p><p>&nbsp;</p><p><span>Select one of the following options that offers the best explanation.</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Calculate slope, intercept using summary statistics@
qu.1.7.comment=<p>To estimate the regression coefficients, we start by estimating the slope, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&beta;</mi><mrow><mn>1</mn></mrow></msub></mrow></mstyle></math>.&nbsp; The estimate for the slope is given by the equation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub></mrow></mfrac></mrow></mstyle></math>.&nbsp; When the appropriate values are substituted in, 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><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$SSxy</mi><mrow><mi>$SSxx</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b1</mi></mrow></mstyle></math>.&nbsp;&nbsp;The&nbsp;estimate the intercept, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&beta;</mi><mrow><mn>0</mn></mrow></msub></mrow></mstyle></math>, is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>Y</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values results in&nbsp;an estimated intercept&nbsp;of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ybar</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$xbar</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b0</mi></mrow></mstyle></math>.&nbsp; Therefore, the estimated least squares regression line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>0</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b0</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi></mrow></mstyle></math>.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$ybar=rand(35,40,4);
$xbar=rand(25,30,4);
$SSxy=rand(100,150,4);
$SSxx=rand(200,250,4);
$b1=$SSxy/$SSxx;
$b0=$ybar-($b1*$xbar);@
qu.1.7.uid=dd4df218-33fa-4706-af06-071a8bddaea9@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=Simple Linear Regression;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.7.weighting=1,1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.answer.units=@
qu.1.7.part.1.numStyle=   @
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.showUnits=false@
qu.1.7.part.1.err=0.01@
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=$b0@
qu.1.7.part.2.name=sro_id_2@
qu.1.7.part.2.answer.units=@
qu.1.7.part.2.numStyle=   @
qu.1.7.part.2.editing=useHTML@
qu.1.7.part.2.showUnits=false@
qu.1.7.part.2.err=0.0010@
qu.1.7.part.2.question=(Unset)@
qu.1.7.part.2.mode=Numeric@
qu.1.7.part.2.grading=toler_abs@
qu.1.7.part.2.negStyle=both@
qu.1.7.part.2.answer.num=$b1@
qu.1.7.question=<p>Measurements on two variables, <em>X</em> and <em>Y</em>, were taken from 15 individuals, with the following summary statistics:</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><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>Y</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ybar</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$SSxy</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSxx</mi></mrow></mstyle></math></p><p>&nbsp;</p><p>Use this information to estimate the parameters in the model<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&beta;</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>&beta;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mi>&epsilon;</mi></mrow></mrow></mstyle></math>, using the method of least squares.</p><p>&nbsp;</p><p>Enter&nbsp;your estimates for the intercept and slope in the appropriate spaces below.</p><p>&nbsp;</p><p>Round each of your values to at least 3 decimal places.</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp; +&nbsp;<span>&nbsp;</span><2><span>&nbsp;<em>X</em></span></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=("'The purpose of linear regression is to investigate if there exists a linear relationship between a response variable and one or more explanatory variables.'");
$b=("'A residual is the difference between an observed and predicted value of Y, for a particular value of X.'");
$c=("'If the coefficient of determination is very large, it would be reasonable to expect that a hypothesis test on the slope would result in a very small p-value.'");
$d=("'A transformation may be applied to the response variable or the explanatory variable (or both), in an attempt to establish a linear relationship between the variables.'");
$e=("'If the purpose of our regression model is prediction, it does not matter which variables we define as the explanatory and response variable.'");
$f=("'The observed values of Y will fall on the estimated regression line, while the predicted values of Y will vary around the regression line.'");
$g=("'The sum of the residuals must be 1.'");
$h=("'If a confidence interval for the true value of the slope contains 1, we can be certain that there is no evidence of a linear relationship between X and Y.'");
$i=("'If a plot of the residuals versus fitted values shows a distinct straight-line (i.e. linear) pattern, then the model assumptions have been verified, and the inference procedures are valid.'");
$jm=maple("
J1:=convert(cat(`The Method of Least Squares is a method of fitting a regression line to the data, such that`,MathML[ExportPresentation](sum(e[i]^2,i=1..n)),`is minimized.`),string):
J1
");
$j=switch(0,$jm);
$Answers=["'True'","'True'","'True'","'True'","'False'","'False'","'False'","'False'","'False'","'True'"];
$Distractors=["'False'","'False'","'False'","'False'","'True'","'True'","'True'","'True'","'True'","'False'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A1=switch($k1, $Answers);
$D1=switch($k1, $Distractors);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A2=switch($k2, $Answers);
$D2=switch($k2, $Distractors);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A3=switch($k3, $Answers);
$D3=switch($k3, $Distractors);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A4=switch($k4, $Answers);
$D4=switch($k4, $Distractors);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$g,$h,$i,"$j");
$A5=switch($k5, $Answers);
$D5=switch($k5, $Distractors);@
qu.1.8.uid=856ef9e6-906f-4caa-84a4-539fe4a9d5a1@
qu.1.8.info=  Course=Introductory Statistics;
  Topic=Simple Linear Regression;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.8.weighting=1,1,1,1,1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.grader=exact@
qu.1.8.part.1.name=sro_id_1@
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.display.permute=true@
qu.1.8.part.1.question=(Unset)@
qu.1.8.part.1.answer.2=$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>@

qu.1.9.mode=Inline@
qu.1.9.name=Calculate test statistic, p-value for two-sided hypothesis test@
qu.1.9.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><msub><mi>&beta;</mi><mrow><mn>1</mn></mrow></msub><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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>H</mi><mrow><mi mathvariant='normal'>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&beta;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>.</p>
<p>b)&nbsp; The test statistic to test the null hypothesis that the true slope is equal to zero is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub></mrow></mfenced></mrow></mfrac></mrow></mrow></mstyle></math>, where&nbsp;the standard error of the estimate of the slope is given by the equation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mi>&beta;</mi><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>s</mi><mrow><msqrt><mrow><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; When the appropriate values are substituted in, this results in&nbsp;a standard error of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msqrt><mrow><mi>$s2</mi></mrow></msqrt><mrow><msqrt><mrow><mi>$SSxx</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SEb1</mi></mrow></mstyle></math>.&nbsp; Using this value to calculate the <em>t</em> test statistic gives us <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$b1</mi><mrow><mi>$SEb1</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; To determine if there is sufficient evidence to reject the null hypothesis, we need to calculate a p-value.&nbsp; Since the alternative hypothesis indicates a two-sided test, the p-value is found as&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><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='0.2777778em' 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&nbsp;<em>t </em>follows a&nbsp;<em>t</em> distribution with <em>n - 2 = $df</em>&nbsp;degrees of freedom.&nbsp;&nbsp;Using computer software, or approximating with&nbsp;a <em>t </em>distribution table, the p-value is found to be $pvalue.&nbsp; Since&nbsp;this value is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is significant evidence that the true slope is not equal to 0.&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$SSxy=rand(1000,1200,5);
$SSxx=rand(800,999,4);
$SSyy=rand(1500,1600,5);
$s2=rand(40,50,4);
$n=range(20,30);
$df=$n-2;
$xbar=rand(150,200,4);
$ybar=rand(450,500,4);
$b1=$SSxy/$SSxx;
$b0=$ybar-($b1*$xbar);
$SEb1=sqrt($s2)/sqrt($SSxx);
$tTest=$b1/$SEb1;
$Tail=1-studentst($df,$tTest);
$pvalue=2*$Tail;
condition:lt($pvalue,0.05);@
qu.1.9.uid=f9302c42-c339-487b-abe7-f4dab1540bd9@
qu.1.9.info=  Course=Introductory Statistics;
  Topic=Simple Linear Regression;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.9.weighting=1,1,1@
qu.1.9.numbering=alpha@
qu.1.9.part.1.name=sro_id_1@
qu.1.9.part.1.editing=useHTML@
qu.1.9.part.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><msub><mi>&beta;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</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><msub><mi>&beta;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>@
qu.1.9.part.1.fixed=@
qu.1.9.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><mrow><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>0</mn></mrow></msub></mrow><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><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>0</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>@
qu.1.9.part.1.question=null@
qu.1.9.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><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><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><msub><mover><mrow><mi>&beta;</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>@
qu.1.9.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><msub><mi>&beta;</mi><mrow><mn>0</mn></mrow></msub><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><msub><mi>&beta;</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>@
qu.1.9.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&beta;</mi><mrow><mn>1</mn></mrow></msub><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><msub><mi>&beta;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>@
qu.1.9.part.1.mode=Multiple Choice@
qu.1.9.part.1.display=vertical@
qu.1.9.part.1.answer=1@
qu.1.9.part.2.name=sro_id_2@
qu.1.9.part.2.answer.units=@
qu.1.9.part.2.numStyle=   @
qu.1.9.part.2.editing=useHTML@
qu.1.9.part.2.showUnits=false@
qu.1.9.part.2.err=0.01@
qu.1.9.part.2.question=(Unset)@
qu.1.9.part.2.mode=Numeric@
qu.1.9.part.2.grading=toler_abs@
qu.1.9.part.2.negStyle=both@
qu.1.9.part.2.answer.num=$tTest@
qu.1.9.part.3.name=sro_id_3@
qu.1.9.part.3.editing=useHTML@
qu.1.9.part.3.choice.5=There is significant evidence that the population mean is different from 0.@
qu.1.9.part.3.fixed=@
qu.1.9.part.3.choice.4=There is no significant evidence that the sample slope is different from 0.@
qu.1.9.part.3.question=null@
qu.1.9.part.3.choice.3=There is significant evidence that the sample slope is different from 0.@
qu.1.9.part.3.choice.2=There is no significant evidence that the true population slope is different from 0.@
qu.1.9.part.3.choice.1=There is significant evidence that the true population slope is different from 0.@
qu.1.9.part.3.mode=Multiple Choice@
qu.1.9.part.3.display=vertical@
qu.1.9.part.3.answer=1@
qu.1.9.question=<p>A random sample of $n individuals is drawn from a population, and measurements on two variables, <em>X</em> and <em>Y,</em>&nbsp;are obtained from each.&nbsp; The following are the summary statistics that are obtained.</p><p>&nbsp;</p><p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mover><mrow><mi>Y</mi></mrow><mi>&macr;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ybar</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>xy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSxy</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>xx</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSxx</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>SS</mi><mrow><mi>yy</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSyy</mi></mrow></mstyle></math></p><p>&nbsp;</p><p>The regression line for examining the linear relationship between <em>X</em> and <em>Y</em> is found to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>Y</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$b0</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>$b1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi></mrow></mstyle></math>.&nbsp; We are interested in testing whether or not the true population slope is different from 0.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What are the appropriate null and alternative hypotheses?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the value of the <em>t</em> 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; What is the appropriate conclusion that can be made, at the 5% level of significance?</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

