qu.1.topic=Inference for Two Population Variances@

qu.1.1.mode=Inline@
qu.1.1.name=Calculate test statistic, range of p-value for two-sided hypothesis test@
qu.1.1.comment=<p>a)&nbsp; Before we can calculate the test statistic, we need to calculate each of the sample variances,&nbsp;using the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate each of the sample variances 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><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s1</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi></mrow></mstyle></math>.&nbsp; The <em>F</em> test statistic is calculated as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the values calculated from part (a), we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>$s1</mi><mrow><mi>$s2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$FStat</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; In order to calculate the p-value, we need to find the area under the <em>F</em> distribution, at 5 degrees of freedom in the numerator and 4 degrees of freedom in the denominator.&nbsp; Since we are conducting a two-sided hypothesis test, the p-value is obtained by determining the area to the right of the test statistic and multplying that value by 2.&nbsp; That is, the p-value is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$FStat</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p>Using computer software, we can find the area under the <em>F</em> distribution, to the right of the test statistic, to be $UpperTail.&nbsp; Multiplying this value by 2 gives us a p-value of $pvalue.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis at the 10% level of significance, in favour of the alternative hypothesis that the population variances are not equal.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$Data1=[rand(50,80,4),rand(50,80,4),rand(50,80,4),rand(50,80,4),rand(50,80,4),rand(50,80,4)];
$Data2=[rand(60,70,4),rand(60,70,4),rand(60,70,4),rand(60,70,4),rand(60,70,4)];
$Values=maple("
S1:=Statistics[Variance]($Data1):
S2:=Statistics[Variance]($Data2):
S1, S2
");
$s1=switch(0, $Values);
$s2=switch(1, $Values);
condition:gt($s1,$s2);
$FStat=$s1/$s2;
$LowerTail=maple("
X:=Statistics[CDF](FRatio(5,4), $FStat):
X
");
$UpperTail=1-$LowerTail;
$pvalue=2*$UpperTail;
condition:lt($pvalue,0.10);@
qu.1.1.uid=a105c43d-6fea-4d15-bcf5-e9ead6ddf352@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Variances, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.1.part.1.err=0.0010@
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qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$FStat@
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qu.1.1.part.2.question=null@
qu.1.1.part.2.choice.3=p-value < 0.10@
qu.1.1.part.2.choice.2=0.10 < p-value < 0.50@
qu.1.1.part.2.choice.1=p-value > 0.50@
qu.1.1.part.2.mode=Non Permuting Multiple Choice@
qu.1.1.part.2.display=vertical@
qu.1.1.part.2.answer=3@
qu.1.1.part.3.name=sro_id_3@
qu.1.1.part.3.editing=useHTML@
qu.1.1.part.3.fixed=@
qu.1.1.part.3.question=null@
qu.1.1.part.3.choice.2=There is insufficicent evidence to reject the null hypothesis at the 10% level of significance, and therefore no significant&nbsp;evidence that the population variances are not equal to each other.@
qu.1.1.part.3.choice.1=There is sufficient evidence to reject the null hypothesis at the 10% level of significance, and therefore evidence that the population variances are not equal to each other.@
qu.1.1.part.3.mode=Multiple Choice@
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qu.1.1.part.3.answer=1@
qu.1.1.question=<p>The following observations are from two independent&nbsp;random samples, drawn from normally distributed populations.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="400" align="center">    <tbody>        <tr>            <td><strong>Sample 1</strong></td>            <td>$Data1</td>        </tr>        <tr>            <td><strong>Sample 2</strong></td>            <td>$Data2</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>Test the null hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mstyle></math>&nbsp;against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&ne;</mo></mrow><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mstyle></math>.</p><p>&nbsp;</p><p><span><span>a)&nbsp; Using the larger sample variance in the numerator, calculate the <em>F</em>&nbsp; test statistic.</span></span></p><p>&nbsp;</p><p><span><span>Round your response to at least 3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><1><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>b)&nbsp; The p-value falls within which one of the following ranges:</span></span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span><span>c)&nbsp; What conclusion can be made at the 10% level of significance?</span></span></span></span></p><p>&nbsp;</p><p><span><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Determine hypotheses, calculate test statistic, range of p-value for one-sided hypothesis test@
qu.1.2.comment=<p>a)&nbsp; The null hypothesis 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>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><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mstyle></math>, against the one-sided alternate hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mstyle></math>, where population 1 is such that it has the larger corresponding sample variance.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Before we can calculate the test statistic, we need to calculate each of the sample variances,&nbsp;using the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate each of the sample variances 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><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s1</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi></mrow></mstyle></math>.&nbsp; Since the second sample has the larger sample variance, we will redefine this to be Population 1.&nbsp; The <em>F</em> test statistic is calculated as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>,&nbsp;where we want to have the larger sample variance in the numerator.&nbsp; Substituting in the values for sample variances, 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>F</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>$s2</mi></mrow><mrow><mi>$s1</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$FStat</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; In order to calculate the p-value, we need to find the area under the <em>F</em> distribution, at&nbsp;6 degrees of freedom in the numerator and&nbsp;4 degrees of freedom in the denominator.&nbsp; Since we are conducting a one-sided, upper-tailed hypothesis test, the p-value is obtained by determining the area to the right of the test statistic.</p>
<p>Using computer software, we can find the a p-value of $pvalue.&nbsp; Since the p-value is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis at the 5% level of significance, in favour of the alternative hypothesis that the variance of Population&nbsp;1 is greater than that of Population 2.</p>@
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qu.1.2.solution=@
qu.1.2.algorithm=$Data1=[rand(5,10,3),rand(5,10,3),rand(5,10,3),rand(5,10,3),rand(5,10,3)];
$Data2=[rand(25,40,4),rand(25,40,4),rand(25,40,4),rand(25,40,4),rand(25,40,4),rand(25,40,4),rand(25,40,4)];
$Values=maple("
S1:=Statistics[Variance]($Data1):
S2:=Statistics[Variance]($Data2):
S1, S2
");
$s1=switch(0, $Values);
$s2=switch(1, $Values);
$FStat=$s2/$s1;
$LowerTail=maple("
X:=Statistics[CDF](FRatio(6,4), $FStat):
X
");
$pvalue=1-$LowerTail;
condition:lt($pvalue,0.05);@
qu.1.2.uid=63467d67-e280-4b63-974f-829d6ada27eb@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Variances, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.2.part.3.choice.2=There is insufficicent evidence to reject the null hypothesis at the 5% level of significance, and therefore no significant evidence that the population variances are not equal to each other.@
qu.1.2.part.3.choice.1=There is sufficient evidence to reject the null hypothesis at the 5% level of significance, and therefore evidence that the variance of Population&nbsp;1 is larger than the variance of Population 2.@
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qu.1.2.question=<p>The following observations are from two independent&nbsp;random samples, drawn from normally distributed populations.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="400" align="center">    <tbody>        <tr>            <td><strong>Sample 1</strong></td>            <td>$Data1</td>        </tr>        <tr>            <td><strong>Sample 2</strong></td>            <td>$Data2</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>We are interested in testing the null hypothesis that the two population variances are equal, against the one-sided alternative that the variance of Population 1&nbsp;is larger than the variance of Population 2.</p><p>&nbsp;</p><p>Define Population 1 to be the population with the larger sample variance.</p><p>&nbsp;</p><p><span><span>a)&nbsp; What are the appropriate null and alternative hypotheses?</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>b)&nbsp;&nbsp;What is the value of the&nbsp;<em>F</em> test statistic?</span></span></span></p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.&nbsp;</p><p>&nbsp;</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span><span>c)&nbsp; What conclusion can be made at the 5% level of significance?</span></span></span></span></p><p>&nbsp;</p><p><span><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></span></p>@

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qu.1.3.name=Determine F values, upper limit for 90% confidence interval@
qu.1.3.comment=<p>a)&nbsp; For a 90% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>, the&nbsp;critical <em>F</em>&nbsp;values are from&nbsp;the<em> F</em> distribution with an area to their right of 0.05 (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><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>).&nbsp; For the lower bound, the&nbsp;<em>F</em> value&nbsp;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>F</mi><mrow><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>F</mi><mrow><mi>$df1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$df2</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$F12</mi></mrow></mstyle></math>.&nbsp; In the upper bound, the&nbsp;<em>F </em>value&nbsp;is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>F</mi><mrow><mi>$df2</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$df1</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$F21</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The formula for a confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>&nbsp;is given&nbsp;as: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>F</mi><mrow><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>F</mi><mrow><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we&nbsp;get a lower limit 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><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$var1</mi><mrow><mi>$var2</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>$F12</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Lower</mi></mrow></mstyle></math>and an upper limit 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><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$var1</mi><mrow><mi>$var2</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$F21</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Upper</mi></mrow></mstyle></math>.&nbsp;&nbsp;Therefore, a 90% confidence interval for the ratio of the population variances is ($Lower, $Upper).&nbsp;</p>@
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qu.1.3.algorithm=$n1=range(7, 10);
$n2=range(6, 9);
$df1=$n1-1;
$df2=$n2-1;
$var1=rand(7, 9, 3);
$var2=rand(3, 5, 3);
$Fvalues=maple("
X:=Statistics[Quantile](FRatio($df1, $df2), 0.95):
Y:=Statistics[Quantile](FRatio($df2, $df1), 0.95):
X, Y
");
$F12=switch(0, $Fvalues);
$F21=switch(1, $Fvalues);
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qu.1.3.uid=b945bcfb-4a1c-4d4f-a1a1-803e7e58fd19@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Variances, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
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qu.1.3.part.3.answer.num=$Upper@
qu.1.3.question=<p>Independent&nbsp;random samples&nbsp;are drawn from two normally distributed&nbsp;populations.&nbsp; From the first population, a random sample of size <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi></mrow></mstyle></math>is obtained, with a sample variance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var1</mi></mrow></mstyle></math>.&nbsp; A random sample of size <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n2</mi></mrow></mstyle></math>is drawn from the second population, with a corresponding sample variance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var2</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Determine the appropriate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub></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>F</mi><mrow><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow></mstyle></math>values required for a 90% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>.</p><p>&nbsp;</p><p>Round your responses to at least 3 decimal places.</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><msub><mi>F</mi><mrow><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>F</mi><mrow><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>b)&nbsp; Calculate the upper bound of the 90% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>.</span></span></p><p>&nbsp;</p><p><span><span>Round your response to at least 3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p>@

qu.1.4.mode=Multiple Selection@
qu.1.4.name=Definitions 2: Inference for Two Population Variances.@
qu.1.4.comment=@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=@
qu.1.4.uid=d83f26ce-d255-4683-8f07-fc61723f5838@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Variances;
  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=In determining a confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>, the F distribution values used to calculate the margin of error may be different for the lower and upper bound.@
qu.1.4.choice.2=If a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&alpha;</mi></mrow></mfenced></mrow></mstyle></math>*100% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math> contains the value 1, then the null hypothesis (with a two-sided alternative) would not be rejected at the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi></mrow></mstyle></math> % level of significance.@
qu.1.4.choice.3=If <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>, we can be sure <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.@
qu.1.4.choice.4=Since the F distribution is asymmetric, it is not possible to carry out a two-sided hypothesis test.@
qu.1.4.fixed=@

qu.1.5.mode=Inline@
qu.1.5.name=Calculate 95% confidence interval, conclusions@
qu.1.5.comment=<p>a)&nbsp; In order to determine the 95% confidence interval for the ratio of the population variances, we first must calculate the sample variances.&nbsp; To calculate each of the sample variances, we can use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate each of the sample variances 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><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var1</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var2</mi></mrow></mstyle></math>.</p>
<p>For a 95% confidence interval, the&nbsp;critical <em>F</em>&nbsp;values are values from the <em>F </em>distribution, at appropriate degrees of freedom, such that the area above these values is 0.025.&nbsp; Using computer software, or approximating from an <em>F</em> distribution table, these values are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>F</mi><mrow><mn>5</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>4</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$F12</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>F</mi><mrow><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>F</mi><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>5</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$F21</mi></mrow></mstyle></math>.</p>
<p>Finally, to calculate the lower and upper bounds of the confidence interval, we use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>F</mi><mrow><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>F</mi><mrow><msub><mi>df</mi><mrow><mn>2</mn></mrow></msub><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><msub><mi>df</mi><mrow><mn>1</mn></mrow></msub></mrow></msub></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we obtain a 95% confidence interval for the ratio of the population variances 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><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$var1</mi><mrow><mi>$var2</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>$F12</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$var1</mi><mrow><mi>$var2</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$F21</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mi>$Lower</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>$Upper</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>b)&nbsp; Because the value 1 is contained within the 95% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>, at the 5% level of significance we would&nbsp;not expect to&nbsp;reject the null hypothesis, in favour of the two-sided alternative.</p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$Data1=[rand(1,9,3),rand(1,9,3),rand(1,9,3),rand(1,9,3),rand(1,9,3),rand(1,9,3)];
$Data2=[rand(5,8,3),rand(5,8,3),rand(5,8,3),rand(5,8,3),rand(5,8,3)];
$Data=maple("
V1:=Statistics[Variance]($Data1):
V2:=Statistics[Variance]($Data2):
F1:=Statistics[Quantile](FRatio(5,4),0.975):
F2:=Statistics[Quantile](FRatio(4,5),0.975):
V1, V2, F1, F2
");
$var1=switch(0, $Data);
$var2=switch(1, $Data);
$F12=switch(2, $Data);
$F21=switch(3,$Data);
condition:gt($var1,$var2);
$Lower=($var1/$var2)*(1/$F12);
$Upper=($var1/$var2)*$F21;
condition:lt($Lower,1.0);@
qu.1.5.uid=aa4c1d22-13c2-4e9e-9566-13a65f91aaf4@
qu.1.5.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Variances, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
qu.1.5.weighting=1,1,1@
qu.1.5.numbering=alpha@
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qu.1.5.part.1.err=0.01@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.mode=Numeric@
qu.1.5.part.1.grading=toler_abs@
qu.1.5.part.1.negStyle=both@
qu.1.5.part.1.answer.num=$Lower@
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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=$Upper@
qu.1.5.part.3.name=sro_id_3@
qu.1.5.part.3.editing=useHTML@
qu.1.5.part.3.fixed=@
qu.1.5.part.3.choice.4=It is impossible to determine, since the F distribution is not a symmetric distribution.@
qu.1.5.part.3.question=null@
qu.1.5.part.3.choice.3=Because the upper limit of the 95% confidence interval is&nbsp;quite large, there is strong evidence against the null hypothesis, and therefore it would be rejected in favour of the alternative hypothesis.@
qu.1.5.part.3.choice.2=Because 0 is not contained within the 95% confidence interval, we would reject the null hypothesis, and therefore have evidence that the population variances are different.@
qu.1.5.part.3.choice.1=Because 1 is contained within the 95% confidence interval, we would not reject the null hypothesis, and therefore have no significant evidence that the population variances are different.@
qu.1.5.part.3.mode=Multiple Choice@
qu.1.5.part.3.display=vertical@
qu.1.5.part.3.answer=1@
qu.1.5.question=<p>The following observations are from two independent&nbsp;random samples, drawn from normally distributed populations.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="400" align="center">    <tbody>        <tr>            <td><strong>Sample 1</strong></td>            <td>$Data1</td>        </tr>        <tr>            <td><strong>Sample 2</strong></td>            <td>$Data2</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Calculate a 95% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>.</p><p>&nbsp;</p><p>Enter your answers for the lower confidence limit and the upper confidence limit separately, in the two separate spaces provided.</p><p>&nbsp;</p><p>Round your responses to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span>&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><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;&nbsp;</p><p><span><span>b)&nbsp; If you were to conduct a hypothesis test of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mstyle></math>versus <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&ne;</mo></mrow><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mstyle></math>&nbsp;at the 5% level of significance, what would be the expected result?</span></span></p><p>&nbsp;</p><p><span><span>Select one of the following options that offers the best explanation.</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.1.6.mode=Multiple Selection@
qu.1.6.name=Definitions 1: Inference for Two Population Variances.@
qu.1.6.comment=@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=@
qu.1.6.uid=f4b51490-7fa9-43d2-90c6-0c59c24ce9f9@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Variances;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.6.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>Note that there may be more than one correct answer; select all that are true.</p>@
qu.1.6.answer=1, 2@
qu.1.6.choice.1=In hypothesis testing for two variances, if the null hypothesis is true, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>.@
qu.1.6.choice.2=If, in hypothesis testing, we designate population 1 to be the population with the larger sample variance, then a one-sided hypothesis test can always be designated 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><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mrow><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mrow></mstyle></math>.@
qu.1.6.choice.3=If the null hypothesis is true, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math> follows an F distribution with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>and</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math> degrees of freedom.@
qu.1.6.choice.4=In order for the inference procedures on two population variances to be valid, the two populations from which the samples are drawn must be normally distributed, but not necessary independent.@
qu.1.6.fixed=@

qu.1.7.mode=Inline@
qu.1.7.name=Calculate sample variances, test statistic@
qu.1.7.comment=<p>a)&nbsp; To calculate each of the sample variances, we can use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate each of the sample variances 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><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s1</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The <em>F</em> test statistic is calculated as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the values calculated from part (a), we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>F</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>$s1</mi><mrow><mi>$s2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$FStat</mi></mrow></mstyle></math>.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$Data1=[rand(0.1,0.99,2),rand(0.1,0.99,2),rand(0.1,0.99,2),rand(0.1,0.99,2),rand(0.1,0.99,2),rand(0.1,0.99,2)];
$Data2=[rand(0.5,0.8,2),rand(0.5,0.8,2),rand(0.5,0.8,2),rand(0.5,0.8,2),rand(0.5,0.8,2)];
$Values=maple("
S1:=Statistics[Variance]($Data1):
S2:=Statistics[Variance]($Data2):
S1, S2
");
$s1=switch(0, $Values);
$s2=switch(1, $Values);
condition:gt($s1,$s2);
$FStat=$s1/$s2;@
qu.1.7.uid=ccbe73b9-c1f0-44f6-8313-a82fded1b03b@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Variances, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.7.weighting=1,1,1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.answer.units=@
qu.1.7.part.1.numStyle=   @
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.showUnits=false@
qu.1.7.part.1.err=0.01@
qu.1.7.part.1.question=(Unset)@
qu.1.7.part.1.mode=Numeric@
qu.1.7.part.1.grading=toler_abs@
qu.1.7.part.1.negStyle=both@
qu.1.7.part.1.answer.num=$s1@
qu.1.7.part.2.name=sro_id_2@
qu.1.7.part.2.answer.units=@
qu.1.7.part.2.numStyle=   @
qu.1.7.part.2.editing=useHTML@
qu.1.7.part.2.showUnits=false@
qu.1.7.part.2.err=0.01@
qu.1.7.part.2.question=(Unset)@
qu.1.7.part.2.mode=Numeric@
qu.1.7.part.2.grading=toler_abs@
qu.1.7.part.2.negStyle=both@
qu.1.7.part.2.answer.num=$s2@
qu.1.7.part.3.name=sro_id_3@
qu.1.7.part.3.answer.units=@
qu.1.7.part.3.numStyle=   @
qu.1.7.part.3.editing=useHTML@
qu.1.7.part.3.showUnits=false@
qu.1.7.part.3.err=0.01@
qu.1.7.part.3.question=(Unset)@
qu.1.7.part.3.mode=Numeric@
qu.1.7.part.3.grading=toler_abs@
qu.1.7.part.3.negStyle=both@
qu.1.7.part.3.answer.num=$FStat@
qu.1.7.question=<p>The following observations are from two independent&nbsp;random samples, drawn from normally distributed populations.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="400" align="center">    <tbody>        <tr>            <td><strong>Sample 1</strong></td>            <td>$Data1</td>        </tr>        <tr>            <td><strong>Sample 2</strong></td>            <td>$Data2</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Calculate the sample variance for each of the samples.</p><p>&nbsp;</p><p>Round your responses to at least 3 decimal places.</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><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>b)&nbsp; Suppose we wish to test the null hypothesis that the population variances are equal.&nbsp; Using the larger sample variance in the numerator, calculate the <em>F</em>&nbsp; test statistic.</span></span></p><p>&nbsp;</p><p><span><span>Round your response to at least 3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

