qu.1.topic=Analysis of Variance@

qu.1.1.mode=Inline@
qu.1.1.name=Definitions 1&2: Random selection of True/False@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$k1=rint(10);
$k2=rint(10);
$k3=rint(10);
$k4=rint(10);
$k5=rint(10);
$z=maple("S := $k1,$k2,$k3,$k4,$k5:
floor( nops({S})/nops([S]) )");
condition: $z;
$a=("'In order for the inferences from ANOVA to be valid, the population variances must be equal.'");
$b=("'The hypothesis testing in ANOVA is always a one-sided, upper tail test.'");
$c=("'If the null hypothesis in ANOVA is rejected, it is usual to carry out pairwise comparisons to determine which population means are different.'");
$d=("'An ANOVA table can contain negative values.'");
$e=("'If the population means are very different, it is likely that the MSE is much greater than the MST.'");
$f=("'If the F test statistic is less than 1, the null hypothesis is automatically rejected.'");
$g=("'The null hypothesis in ANOVA is that the population means are all the same.  Therefore, the appropriate alternative hypothesis is that all the population means are different.'");
$h=("'If the null hypothesis is true, the F test statistic follows an F distribution with k and n degrees of freedom, where k is the number of populations and n is the number of observations.'");
$i=("'ANOVA is only valid when there are more than two populations.  When there are only two groups being examined, you must use a t test.'");
$jm=maple("
J1:=convert(cat(`The MSE in ANOVA is an extended version of the`,MathML[ExportPresentation](s[p]^2),`seen in the pooled-variance t procedure, to account for more than just 2 groups.`),string):
J1
");
$j=switch(0,$jm);
$Answers=["'True'","'True'","'True'","'False'","'False'","'False'","'False'","'False'","'False'","'True'"];
$Distractors=["'False'","'False'","'False'","'True'","'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, $Distractors);@
qu.1.1.uid=273e8e08-b6c0-410a-82a7-ac7cce6ce440@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=One-Way ANOVA;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.1.weighting=1,1,1,1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.grader=exact@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.display.permute=true@
qu.1.1.part.1.question=(Unset)@
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qu.1.1.part.1.answer.1=$A1@
qu.1.1.part.1.mode=List@
qu.1.1.part.1.display=menu@
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qu.1.1.part.1.credit.1=1.0@
qu.1.1.part.2.grader=exact@
qu.1.1.part.2.name=sro_id_2@
qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.display.permute=true@
qu.1.1.part.2.question=(Unset)@
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qu.1.1.part.2.answer.1=$A2@
qu.1.1.part.2.mode=List@
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qu.1.1.part.2.credit.1=1.0@
qu.1.1.part.3.grader=exact@
qu.1.1.part.3.name=sro_id_3@
qu.1.1.part.3.editing=useHTML@
qu.1.1.part.3.display.permute=true@
qu.1.1.part.3.question=(Unset)@
qu.1.1.part.3.answer.2=$D3@
qu.1.1.part.3.answer.1=$A3@
qu.1.1.part.3.mode=List@
qu.1.1.part.3.display=menu@
qu.1.1.part.3.credit.2=0.0@
qu.1.1.part.3.credit.1=1.0@
qu.1.1.part.4.grader=exact@
qu.1.1.part.4.name=sro_id_4@
qu.1.1.part.4.editing=useHTML@
qu.1.1.part.4.display.permute=true@
qu.1.1.part.4.question=(Unset)@
qu.1.1.part.4.answer.2=$D4@
qu.1.1.part.4.answer.1=$A4@
qu.1.1.part.4.mode=List@
qu.1.1.part.4.display=menu@
qu.1.1.part.4.credit.2=0.0@
qu.1.1.part.4.credit.1=1.0@
qu.1.1.part.5.grader=exact@
qu.1.1.part.5.name=sro_id_5@
qu.1.1.part.5.editing=useHTML@
qu.1.1.part.5.display.permute=true@
qu.1.1.part.5.question=(Unset)@
qu.1.1.part.5.answer.2=$D5@
qu.1.1.part.5.answer.1=$A5@
qu.1.1.part.5.mode=List@
qu.1.1.part.5.display=menu@
qu.1.1.part.5.credit.2=0.0@
qu.1.1.part.5.credit.1=1.0@
qu.1.1.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.1.2.mode=Inline@
qu.1.2.name=Complete ANOVA table, interpret results@
qu.1.2.comment=<p>a)&nbsp; The completed ANOVA table is as follows:</p>
<p>&nbsp;</p>
<p>
<table border="1" cellspacing="1" cellpadding="1" width="600" align="center">
    <tbody>
        <tr>
            <td>
            <p align="center"><strong>Source</strong></p>
            </td>
            <td>
            <p align="center"><strong>DF</strong></p>
            </td>
            <td>
            <p align="center"><strong>SS</strong></p>
            </td>
            <td>
            <p align="center"><strong>MS</strong></p>
            </td>
            <td>
            <p align="center"><strong>F</strong></p>
            </td>
        </tr>
        <tr>
            <td>
            <p align="center">Groups</p>
            </td>
            <td>
            <p align="center"><em>$k - 1 = $dfGroup</em></p>
            </td>
            <td>
            <p align="center">$SSG</p>
            </td>
            <td>
            <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><mfrac><mi>$SSG</mi><mrow><mi>$dfGroup</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$MST</mi></mrow></mstyle></math></p>
            </td>
            <td>
            <p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>$MST</mi><mrow><mi>$MSE</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$FStat</mi></mrow></mstyle></math></p>
            </td>
        </tr>
        <tr>
            <td>
            <p align="center">Error</p>
            </td>
            <td>
            <p align="center"><em>$N - $k = $dfError</em></p>
            </td>
            <td>
            <p align="center"><em>$SST - $SSG = $SSE</em></p>
            </td>
            <td>
            <p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>$SSE</mi><mrow><mi>$dfError</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$MSE</mi></mrow></mstyle></math></p>
            </td>
            <td>
            <p align="center">----</p>
            </td>
        </tr>
        <tr>
            <td>
            <p align="center">Total</p>
            </td>
            <td>
            <p align="center"><em>$N - 1 = $dfTotal</em></p>
            </td>
            <td>
            <p align="center">$SST</p>
            </td>
            <td>
            <p align="center">----</p>
            </td>
            <td>
            <p align="center">----</p>
            </td>
        </tr>
    </tbody>
</table>
</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>b)&nbsp; To determine whether or not there is sufficient evidence to reject the null hypothesis, we need to calculate the p-value.&nbsp; The p-value is the area under the <em>F</em> distribution, with $dfGroup and $dfError degrees of freedom, to the right of the test statistic.&nbsp; Using computer software, we can find this area to be exactly <em>p-value = $pvalue.&nbsp; </em>Since this value is very small (i.e. less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>), there is sufficient evidence to reject the null hypothesis, and conclude that at least one of the treatment means is different.</p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$i=rint(3);
$k=switch($i, 3,4,5);
$N=switch($i, 30,44,60);
$dfGroup=$k-1;
$dfError=$N-$k;
$dfTotal=$N-1;
$SSG=rand(800,900,4);
$SST=rand(1100,1200,5);
$SSE=$SST-$SSG;
$MST=$SSG/$dfGroup;
$MSE=$SSE/$dfError;
$FStat=$MST/$MSE;
$Tail=maple("
X:=Statistics[CDF](FRatio($dfGroup, $dfError), $FStat):
X
");
$pvalue=1-$Tail;@
qu.1.2.uid=24b18da3-c74c-4e17-85d8-dd81518bb314@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=One-Way ANOVA;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=ANOVA Table;
  Type=Calculation;
@
qu.1.2.weighting=1,1,1,1,1,1,1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.answer.units=@
qu.1.2.part.1.numStyle=   @
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.showUnits=false@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.mode=Numeric@
qu.1.2.part.1.grading=exact_value@
qu.1.2.part.1.negStyle=both@
qu.1.2.part.1.answer.num=$dfGroup@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.answer.units=@
qu.1.2.part.2.numStyle=   @
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.showUnits=false@
qu.1.2.part.2.err=0.0010@
qu.1.2.part.2.question=(Unset)@
qu.1.2.part.2.mode=Numeric@
qu.1.2.part.2.grading=toler_abs@
qu.1.2.part.2.negStyle=both@
qu.1.2.part.2.answer.num=$MST@
qu.1.2.part.3.name=sro_id_3@
qu.1.2.part.3.answer.units=@
qu.1.2.part.3.numStyle=   @
qu.1.2.part.3.editing=useHTML@
qu.1.2.part.3.showUnits=false@
qu.1.2.part.3.err=0.01@
qu.1.2.part.3.question=(Unset)@
qu.1.2.part.3.mode=Numeric@
qu.1.2.part.3.grading=toler_abs@
qu.1.2.part.3.negStyle=both@
qu.1.2.part.3.answer.num=$FStat@
qu.1.2.part.4.name=sro_id_4@
qu.1.2.part.4.answer.units=@
qu.1.2.part.4.numStyle=   @
qu.1.2.part.4.editing=useHTML@
qu.1.2.part.4.showUnits=false@
qu.1.2.part.4.question=(Unset)@
qu.1.2.part.4.mode=Numeric@
qu.1.2.part.4.grading=exact_value@
qu.1.2.part.4.negStyle=both@
qu.1.2.part.4.answer.num=$dfError@
qu.1.2.part.5.name=sro_id_5@
qu.1.2.part.5.answer.units=@
qu.1.2.part.5.numStyle=   @
qu.1.2.part.5.editing=useHTML@
qu.1.2.part.5.showUnits=false@
qu.1.2.part.5.err=0.0010@
qu.1.2.part.5.question=(Unset)@
qu.1.2.part.5.mode=Numeric@
qu.1.2.part.5.grading=toler_abs@
qu.1.2.part.5.negStyle=both@
qu.1.2.part.5.answer.num=$SSE@
qu.1.2.part.6.name=sro_id_6@
qu.1.2.part.6.answer.units=@
qu.1.2.part.6.numStyle=   @
qu.1.2.part.6.editing=useHTML@
qu.1.2.part.6.showUnits=false@
qu.1.2.part.6.err=0.0010@
qu.1.2.part.6.question=(Unset)@
qu.1.2.part.6.mode=Numeric@
qu.1.2.part.6.grading=toler_abs@
qu.1.2.part.6.negStyle=both@
qu.1.2.part.6.answer.num=$MSE@
qu.1.2.part.7.name=sro_id_7@
qu.1.2.part.7.answer.units=@
qu.1.2.part.7.numStyle=   @
qu.1.2.part.7.editing=useHTML@
qu.1.2.part.7.showUnits=false@
qu.1.2.part.7.question=(Unset)@
qu.1.2.part.7.mode=Numeric@
qu.1.2.part.7.grading=exact_value@
qu.1.2.part.7.negStyle=both@
qu.1.2.part.7.answer.num=$dfTotal@
qu.1.2.part.8.name=sro_id_8@
qu.1.2.part.8.editing=useHTML@
qu.1.2.part.8.fixed=@
qu.1.2.part.8.question=null@
qu.1.2.part.8.choice.2=No, there is not enough evidence to reject the null hypothesis.@
qu.1.2.part.8.choice.1=Yes, there is enough evidence to reject the null hypothesis.@
qu.1.2.part.8.mode=Multiple Choice@
qu.1.2.part.8.display=vertical@
qu.1.2.part.8.answer=1@
qu.1.2.question=<p>A random sample of $N individuals is selected from a population, and these individuals are then randomly assigned to $k treatment groups (such that each group has an equal number of individuals).</p><p>The following partially-completed ANOVA table summarizes the results of measurements taken from each individual, within each group, on a characteristic of interest.</p><p>Use the information to test the null hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&mu;</mi><mrow><mi>$k</mi></mrow></msub></mrow></mstyle></math>, versus 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><mi>At</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>least</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>one</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>mean</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>is</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>different</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; Complete the ANOVA table.</p><p>&nbsp;</p><p>Round any decimal numbers to at least&nbsp;3 decimal places before entering them in the table.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="450" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Source</strong></p>            </td>            <td>            <p align="center"><strong>DF</strong></p>            </td>            <td>            <p align="center"><strong>SS</strong></p>            </td>            <td>            <p align="center"><strong>MS</strong></p>            </td>            <td>            <p align="center"><strong>F</strong></p>            </td>        </tr>        <tr>            <td>            <p align="center">Groups</p>            </td>            <td>            <p align="center"><span>&nbsp;</span><1><span>&nbsp;</span></p>            </td>            <td>            <p align="center">$SSG</p>            </td>            <td>            <p align="center"><span>&nbsp;</span><2><span>&nbsp;</span></p>            </td>            <td>            <p align="center"><span>&nbsp;</span><3><span>&nbsp;</span></p>            </td>        </tr>        <tr>            <td>            <p align="center">Error</p>            </td>            <td>            <p align="center"><span>&nbsp;</span><4><span>&nbsp;</span></p>            </td>            <td>            <p align="center"><span>&nbsp;</span><5><span>&nbsp;</span></p>            </td>            <td>            <p align="center"><span>&nbsp;</span><6><span>&nbsp;</span></p>            </td>            <td>            <p align="center">---</p>            </td>        </tr>        <tr>            <td>            <p align="center">Total</p>            </td>            <td>            <p align="center"><span>&nbsp;</span><7><span>&nbsp;</span></p>            </td>            <td>            <p align="center">$SST</p>            </td>            <td>            <p align="center">---</p>            </td>            <td>            <p align="center">---</p>            </td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>b)&nbsp; At the 5% level of significance, is there sufficient evidence to reject the null hypothesis, and conclude that at least one of the treatment means is different?</p><p>&nbsp;</p><p><span>&nbsp;</span><8><span>&nbsp;</span></p>@

qu.1.3.mode=Multiple Selection@
qu.1.3.name=Definitions 1: Inference for ANOVA@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=@
qu.1.3.uid=154a52b6-39c3-4c94-ae11-c39355c8597e@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=One-Way ANOVA;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.3.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.1.3.answer=1, 2@
qu.1.3.choice.1=In order for the inferences from ANOVA to be valid, the population variances must be equal.@
qu.1.3.choice.2=The hypothesis testing in ANOVA is always a one-sided, upper tail test.@
qu.1.3.choice.3=An ANOVA table can contain negative values.@
qu.1.3.choice.4=If the population means are very different, it is likely that the MSE is much greater than the MST.@
qu.1.3.choice.5=If the F test statistic is less than 1, the null hypothesis is automatically rejected.@
qu.1.3.fixed=@

qu.1.4.mode=Inline@
qu.1.4.name=Calculate SE, degrees of freedom, and differences for pairwise comparison@
qu.1.4.comment=<p>a)&nbsp; To calculate the standard error for 95% confidence intervals using the LSD method, we can use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>s</mi><mrow><mi>p</mi></mrow></msub><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mi>j</mi></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></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>s</mi><mrow><mi>p</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi>MSE</mi></mrow></msqrt></mrow></mrow></mstyle></math>, which can be obtained from the given ANOVA table.&nbsp; Note that because there are the same number of individuals in each group, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>i</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>n</mi><mrow><mi>j</mi></mrow></msub></mrow></mstyle></math>for all values of <em>i</em>&nbsp;and <em>j</em>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$MSE</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi>$n</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SE</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>b)&nbsp; The appropriate degrees of freedom for&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></mstyle></math> for the 95% confidence intervals using the LSD method is <em>$N -&nbsp;3 = $dfError</em>, the degrees of freedom for error in the ANOVA table.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The 95% confidence intervals for pairwise comparisons, using the LSD method, are 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><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mi>j</mi></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plusmn;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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>X</mi></mrow><mi>&macr;</mi></mover><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mstyle></math>.&nbsp; Using the value for the standard error found in part (a), 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>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tAlpha2</mi></mrow></mstyle></math>, we can find the lower and upper limits of the confidence intervals:</p>
<p>Group 1 - Group 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><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plusmn;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SE</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$LL12</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$UL12</mi></mrow></mfenced></mrow></mstyle></math></p>
<p>Group 1 - Group 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><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xbar3</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plusmn;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SE</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$LL13</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$UL13</mi></mrow></mfenced></mrow></mstyle></math></p>
<p>Group 2 - Group 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><mfenced open='(' close=')' separators=','><mrow><mi>$xbar2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xbar3</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plusmn;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SE</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$LL23</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$UL23</mi></mrow></mfenced></mrow></mstyle></math></p>
<p>The confidence interval for Group 1 - Group 2, and Group 2 - Group 3 contains 0, therefore there is no significant difference between Groups 1 and 2, and between Groups 2 and 3.&nbsp; However, the confidence interval for Group 1 - Group 3 does not contain 0, therefore there is evidence of a significant difference between Group 1 and Group 3.</p>@
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$n=$N/3;
$dfError=$N-3;
$dfTotal=$N-1;
$xbar1=rand(20,25,4);
$xbar2=rand(15,20,4);
$xbar3=rand(10,15,4);
$SSG=rand(2000,2200,6);
$SSE=rand(3500,3700,6);
$SST=$SSG+$SSE;
$MSG=$SSG/2;
$MSE=$SSE/$dfError;
$FTest=$MSG/$MSE;
$Tail=maple("
X:=Statistics[CDF](FRatio(2,$dfError), $FTest):
X
");
$pvalue=1-$Tail;
$SE=sqrt($MSE*(1/$n + 1/$n));
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$LL12=($xbar1-$xbar2)-($tAlpha2*$SE);
$UL12=($xbar1-$xbar2)+($tAlpha2*$SE);
$LL13=($xbar1-$xbar3)-($tAlpha2*$SE);
$UL13=($xbar1-$xbar3)+($tAlpha2*$SE);
$LL23=($xbar2-$xbar3)-($tAlpha2*$SE);
$UL23=($xbar2-$xbar3)+($tAlpha2*$SE);
condition:lt($LL12,0);
condition:gt($UL12,0);
condition:gt($LL13,0);
condition:gt($UL13,0);
condition:lt($LL23,0);
condition:gt($UL23,0);@
qu.1.4.uid=8f76d8a0-a41e-45e8-8b79-849dc83f5087@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=One-Way ANOVA;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=ANOVA Table;
  Type=Calculation;
@
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qu.1.4.question=<p>A random sample of $N individuals is selected from a population, and these individuals are then randomly assigned to 3 treatment groups (such that each group has an equal number of individuals).</p><p>The following ANOVA table summarizes the results of measurements taken from each individual, within each group, on a characteristic of interest.&nbsp; Due to the small p-value, the null hypothesis is rejected, and at least one of the treatments means is different.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="450" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Source</strong></p>            </td>            <td>            <p align="center"><strong>DF</strong></p>            </td>            <td>            <p align="center"><strong>SS</strong></p>            </td>            <td>            <p align="center"><strong>MS</strong></p>            </td>            <td>            <p align="center"><strong>F</strong></p>            </td>            <td>            <p align="center"><strong>p-value</strong></p>            </td>        </tr>        <tr>            <td>            <p align="center">Groups</p>            </td>            <td>            <p align="center"><span>&nbsp;2</span></p>            </td>            <td>            <p align="center">$SSG</p>            </td>            <td>            <p align="center"><span>&nbsp;$MSG</span><span>&nbsp;</span></p>            </td>            <td>            <p align="center"><span>&nbsp;$FTest</span></p>            </td>            <td>            <p align="center">$pvalue</p>            </td>        </tr>        <tr>            <td>            <p align="center">Error</p>            </td>            <td>            <p align="center"><span>&nbsp;$dfError</span><span>&nbsp;</span></p>            </td>            <td>            <p align="center"><span>&nbsp;$SSE</span></p>            </td>            <td>            <p align="center"><span>&nbsp;$MSE</span></p>            </td>            <td>            <p align="center">---</p>            </td>            <td>&nbsp;</td>        </tr>        <tr>            <td>            <p align="center">Total</p>            </td>            <td>            <p align="center"><span>&nbsp;$dfTotal</span><span>&nbsp;</span></p>            </td>            <td>            <p align="center">$SST</p>            </td>            <td>            <p align="center">---</p>            </td>            <td>            <p align="center">---</p>            </td>            <td>&nbsp;</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>The sample means for each of the treatment groups were 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><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar3</mi></mrow></mstyle></math>.</p><p>Use&nbsp;Fisher's Least Significant Difference (LSD) method to carryout pairwise comparisons between the treatment means, and determine which means are different.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the standard error of the difference in treatment means, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>X</mi></mrow><mi>&macr;</mi></mover><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mi>j</mi></mrow></msub></mrow></mfenced></mrow></mstyle></math>?</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>b)&nbsp;&nbsp;What are the appropriate degrees of freedom for the <em>t</em>&nbsp;distribution used to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>?&nbsp;</p><p><span>&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Determine whether or not there is significant&nbsp;evidence of a difference between each pair of treatment means, at the 5% level of significance:</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="300" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Comparison</strong></p>            </td>            <td>            <p align="center"><strong>Significant Difference</strong></p>            </td>        </tr>        <tr>            <td>            <p align="center">Group 1 - Group 2</p>            </td>            <td>            <p align="center"><span>&nbsp;</span><3><span>&nbsp;</span></p>            </td>        </tr>        <tr>            <td>            <p align="center">Group 1 - Group 3</p>            </td>            <td>            <p align="center"><span>&nbsp;</span><4><span>&nbsp;</span></p>            </td>        </tr>        <tr>            <td>            <p align="center">Group 2 - Group 3</p>            </td>            <td>            <p align="center"><span>&nbsp;</span><5><span>&nbsp;</span></p>            </td>        </tr>    </tbody></table></p>@

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qu.1.5.name=Calculate MSE, F statistic, conclusion@
qu.1.5.comment=<p>a)&nbsp; The mean square error (MSE) is calculated with 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>MSE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>SSE</mi><mrow><msub><mi>df</mi><mrow><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>error</mo></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</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><mn>4</mn></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msubsup><mi>s</mi><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mi>N</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>k</mi></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values results in an MSE 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>MSE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var3</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><mfenced open='(' close=')' separators=','><mrow><mi>$n4</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var4</mi></mrow></mfenced></mrow><mrow><mi>$N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>4</mn></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$MSE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Using the given SST value, along with the MSE value found in part (a), we can complete the ANOVA table as follows:</p>
<p>&nbsp;</p>
<p>
<table border="1" cellspacing="1" cellpadding="1" width="400" align="center">
    <tbody>
        <tr>
            <td>
            <p align="center"><strong>Source</strong></p>
            </td>
            <td>
            <p align="center"><strong>DF</strong></p>
            </td>
            <td>
            <p align="center"><strong>SS</strong></p>
            </td>
            <td>
            <p align="center"><strong>MS</strong></p>
            </td>
            <td>
            <p align="center"><strong>F</strong></p>
            </td>
        </tr>
        <tr>
            <td>
            <p align="center"><strong>Groups</strong></p>
            </td>
            <td>
            <p align="center">3</p>
            </td>
            <td>
            <p align="center">$SSG</p>
            </td>
            <td>
            <p align="center">$MSG</p>
            </td>
            <td>
            <p align="center">$FStat</p>
            </td>
        </tr>
        <tr>
            <td>
            <p align="center"><strong>Error</strong></p>
            </td>
            <td>
            <p align="center">$dfE</p>
            </td>
            <td>
            <p align="center">$SSE</p>
            </td>
            <td>
            <p align="center">$MSE</p>
            </td>
            <td>
            <p align="center">----</p>
            </td>
        </tr>
        <tr>
            <td>
            <p align="center"><strong>Total</strong></p>
            </td>
            <td>
            <p align="center">$dfT</p>
            </td>
            <td>
            <p align="center">$SST</p>
            </td>
            <td>
            <p align="center">----</p>
            </td>
            <td>
            <p align="center">----</p>
            </td>
        </tr>
    </tbody>
</table>
</p>
<p>&nbsp;</p>
<p>Recall that if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>MSE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>SSE</mi><mrow><msub><mi>df</mi><mrow><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>error</mo></mrow></msub></mrow></mfrac></mrow></mrow></mstyle></math>, then <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SSE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>MSE</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>df</mi><mrow><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>error</mo></mrow></msub></mrow></mstyle></math>.&nbsp; Therefore, we could find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SSE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$MSE</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$dfE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSE</mi></mrow></mstyle></math>; using this, we can then find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SSG</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>SST</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>SSE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SST</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$SSE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SSG</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>&nbsp;From this point, the remainder of the ANOVA table can be easily computed.</p>
<p>&nbsp;</p>
<p>c)&nbsp; To determine whether or not there is sufficient evidence to reject the null hypothesis, we need to determine the p-value.&nbsp; In ANOVA, the p-value is the area under an <em>F</em> distribution, with $dfG and $dfE degrees of freedom, to the right of the test statistic.&nbsp; Using computer software, or approximating with an <em>F</em> distribution table, the p-value is found to be $pvalue.&nbsp; As this value is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis at the 5% level of significance.</p>@
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$n2=range(7,10);
$n3=range(8,11);
$n4=range(6,9);
$N=$n1+$n2+$n3+$n4;
$dfG=3;
$dfE=$N-4;
$dfT=$N-1;
$xbar1=rand(25,30,4);
$xbar2=rand(41,46,4);
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$SSE=$MSE*$dfE;
$SST=rand(1200,1300,6);
$SSG=$SST - $SSE;
$MSG=$SSG/$dfG;
$FStat=$MSG/$MSE;
$Tail=maple("
X:=Statistics[CDF](FRatio($dfG, $dfE), $FStat):
X
");
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qu.1.5.info=  Course=Introductory Statistics;
  Topic=One-Way ANOVA;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=ANOVA table;
  Type=Calculation;
@
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qu.1.5.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence to indicate the population means are not equal.@
qu.1.5.part.3.choice.1=There is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis that at least one population mean is different.@
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qu.1.5.question=<p>Researchers are interested in investigating the effect of&nbsp;four different treatments on a particular characteristic.&nbsp;&nbsp;Random samples of individuals are&nbsp;drawn, and assigned to various treatment groups.&nbsp; The table below&nbsp;displays the summary statistics obtained&nbsp;upon the conclusion of the experiment.</p><p>&nbsp;</p><p align="center"><table border="1" cellspacing="1" cellpadding="1" width="300" align="center">    <tbody>        <tr>            <td><strong>Group 1</strong></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var1</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><strong>Group 2</strong></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>$var2</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><strong>Group 3</strong></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var3</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar3</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n3</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><strong>Group 4</strong></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>4</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var4</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>4</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar4</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>4</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n4</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td colspan="4"><strong>SSTotal = </strong>$SST</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>Use this information to 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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&mu;</mi><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>4</mn></mrow></msub></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><mi>At</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>least</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>one</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&mu;</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>is</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>different</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; Calculate the value of MSE.</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 value of the <em>F</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 conclusion 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>@

qu.1.6.mode=Multiple Selection@
qu.1.6.name=Definitions 2: Inference for ANOVA@
qu.1.6.comment=@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=@
qu.1.6.uid=560b8e0f-ea0a-420e-b3d3-e0414f1f5cb0@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=One-Way ANOVA;
  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=If the null hypothesis in ANOVA is rejected, it is usual to carry out pairwise comparisons to determine which population means are different.@
qu.1.6.choice.2=The MSE in ANOVA is an extended version of the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math> seen in the pooled-variance t procedure, to account for more than just two groups.@
qu.1.6.choice.3=The null hypothesis in ANOVA is that the population means are all the same.  Therefore, the appropriate alternative hypothesis is that all the population means are different.@
qu.1.6.choice.4=If the null hypothesis is true, the F test statistic follows an F distribution with k and n degrees of freedom, where k is the number of populations and n is the number of observations.@
qu.1.6.choice.5=ANOVA is only valid when there are more than two populations.  When there are only two groups being examined, you must use a t test.@
qu.1.6.fixed=@

qu.1.7.mode=Inline@
qu.1.7.name=Calculate sp^2, SE for pairwise comparisons@
qu.1.7.comment=<p>a)&nbsp; The&nbsp;pooled variance&nbsp;is calculated with 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><msubsup><mi>s</mi><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>MSE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>SSE</mi><mrow><msub><mi>df</mi><mrow><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>error</mo></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><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><mn>4</mn></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msubsup><mi>s</mi><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mi>N</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>k</mi></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values results in a pooled variance&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><msubsup><mi>s</mi><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n3</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var3</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><mfenced open='(' close=')' separators=','><mrow><mi>$n4</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var4</mi></mrow></mfenced></mrow><mrow><mi>$N</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>4</mn></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$MSE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The margin of error for 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><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>3</mn></mrow></msub></mrow></mrow></mstyle></math>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>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>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>3</mn></mrow></msub></mrow></mfenced></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><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>3</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>s</mi><mrow><mi>p</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></mrow></mfrac></mrow></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><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>3</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$MSE</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>$n2</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mi>$n3</mi></mrow></mfrac></mrow></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SE23</mi></mrow></mstyle></math>.&nbsp; For a <em>t</em> distribution with $dfE degrees of freedom, the corresponding <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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></mstyle></math>&nbsp;value is $tAlpha2, and therefore the margin of error is calculated as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SE23</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME23</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; In order to determine if there is evidence of a significant difference between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi mathvariant='normal'>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>&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><mi>&mu;</mi><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math>, we need to calculate the upper and lower bounds of 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><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>3</mn></mrow></msub></mrow></mrow></mstyle></math>, which is calculated&nbsp;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><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>3</mn></mrow></msub></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&plusmn;</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>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>3</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$xbar2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar3</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plusmn;</mo><mi>$ME23</mi></mrow></mstyle></math>.&nbsp; Therefore, the 95% confidence interval for the difference between population means is <em>($Lower, $Upper)</em>.&nbsp; Since this confidence interval does not contain 0, there is significant evidence of a difference between the means for Group 2 and Group 3.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$n1=range(6,9);
$n2=range(7,10);
$n3=range(8,11);
$n4=range(6,9);
$N=$n1+$n2+$n3+$n4;
$dfG=3;
$dfE=$N-4;
$dfT=$N-1;
$xbar1=rand(25,30,4);
$xbar2=rand(41,46,4);
$xbar3=rand(50,55,4);
$xbar4=rand(27,32,4);
$var1=rand(15,20,3);
$var2=rand(5,10,3);
$var3=rand(25,30,3);
$var4=rand(20,25,3);
$MSE=(($n1-1)*$var1 + ($n2-1)*$var2 + ($n3-1)*$var3 + ($n4-1)*$var4)/(($n1+$n2+$n3+$n4)-4);
$sp=sqrt($MSE);
$PE23=$xbar2-$xbar3;
$SE23=$sp*sqrt(1/$n2 + 1/$n3);
$tAlpha2=invstudentst($dfE, 0.975);
$ME23=$tAlpha2*$SE23;
$Lower=$PE23 - $ME23;
$Upper=$PE23 + $ME23;
condition:lt($Lower,0);
condition:lt($Upper,0);@
qu.1.7.uid=15270de6-a0f4-4eb4-84b6-91e86a10a2b5@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=One-Way ANOVA;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=ANOVA table;
  Type=Calculation;
@
qu.1.7.weighting=1,1,1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.answer.units=@
qu.1.7.part.1.numStyle=   @
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.showUnits=false@
qu.1.7.part.1.err=0.01@
qu.1.7.part.1.question=(Unset)@
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qu.1.7.part.1.grading=toler_abs@
qu.1.7.part.1.negStyle=both@
qu.1.7.part.1.answer.num=$MSE@
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=$ME23@
qu.1.7.part.3.grader=exact@
qu.1.7.part.3.name=sro_id_3@
qu.1.7.part.3.editing=useHTML@
qu.1.7.part.3.display.permute=true@
qu.1.7.part.3.question=(Unset)@
qu.1.7.part.3.answer.2=No, there is no significant evidence of a difference.@
qu.1.7.part.3.answer.1=Yes, there is significant evidence of a difference.@
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qu.1.7.question=<p>Researchers are interested in investigating the effect of&nbsp;four different treatments on a certain characteristic of interest.&nbsp;&nbsp;Random samples of individuals are&nbsp;drawn and assigned to the&nbsp;various treatment groups.&nbsp; The table below&nbsp;displays the summary statistics obtained&nbsp;upon the conclusion of the experiment.</p><p>&nbsp;</p><p align="center"><table border="1" cellspacing="1" cellpadding="1" width="300" align="center">    <tbody>        <tr>            <td><strong>Group 1</strong></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var1</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><strong>Group 2</strong></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>$var2</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><strong>Group 3</strong></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var3</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar3</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n3</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><strong>Group 4</strong></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>4</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var4</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>4</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar4</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>4</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n4</mi></mrow></mstyle></math></td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>A one-way ANOVA was carried out to 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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>&mu;</mi><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>4</mn></mrow></msub></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><mi>At</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>least</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>one</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&mu;</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>is</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>different</mi></mrow></mstyle></math>, with the resulting p-value being small enough to reject the null hypothesis.</p><p>&nbsp;</p><p>Using Fisher's Least Significant Difference (LSD) method, determine if there is a significant difference between Group 2 and Group 3, based on the calculation of a 95% confidence interval.</p><p>&nbsp;</p><p>a)&nbsp; Calculate the value 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><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>.</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; Calculate the margin of error for a 95%&nbsp;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><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>3</mn></mrow></msub></mrow></mrow></mstyle></math>, using&nbsp;Fisher's LSD&nbsp;method.</span></p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><2><span>&nbsp;</span>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Based on the confidence interval calculated in part (b), is there significant&nbsp;evidence of a difference between <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi mathvariant='normal'>&mu;</mi><mrow><mn>2</mn></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>&mu;</mi><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math>?</span></span></p><p>&nbsp;</p><p><span>&nbsp;</span><3><span>&nbsp;</span></p>@

