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

qu.1.1.mode=Multiple Selection@
qu.1.1.name=Definitions 2: Inference for Two Population Means@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=@
qu.1.1.uid=43a9b075-f68d-4e7f-a564-3c6f9f6be1f6@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.1.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.1.1.answer=1, 2@
qu.1.1.choice.1=If the two populations are not normally distributed, then the inference procedures for the difference between two means can still be valid, provided the sample sizes are large.@
qu.1.1.choice.2=The sampling distribution of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mrow></mstyle></math> is approximately normal, provided the independent sampling distributions of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><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><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math> are themselves approximately normal.@
qu.1.1.choice.3=One problem with taking observations on the same experimental unit, as in a paired difference test, is that it can increase the variability in the experiment.@
qu.1.1.choice.4=The results obtained from the pooled-variance t procedure versus the Welch procedure, when applied to the same data set, will always be significantly different.@
qu.1.1.choice.5=The paired difference procedure is less susceptible to the populations being non-normal, since taking the difference between two non-normal populations will result in a normal distribution.@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Selection@
qu.1.2.name=Definitions 1: Inference for Two Population Means@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=@
qu.1.2.uid=e1f8450a-cbde-43e9-91b9-9b1a28a35c4c@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.2.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>Note that there may be more than one correct answer; select all that are true.</p>@
qu.1.2.answer=1, 2, 3@
qu.1.2.choice.1=When applying the paired difference procedure, it is assumed that the differences between pairs of observations constitute a simple random sample from the population of differences.@
qu.1.2.choice.2=The pooled sample variance used in a pooled-variance t procedure is a weighted average of the sample variances, and tends to be closer to the sample variance with the higher number of observations.@
qu.1.2.choice.3=If the sample sizes are similar, the pooled-variance t procedure will still work relatively well, even if the population variances are not quite the same.@
qu.1.2.choice.4=The pooled-variance t procedure requires that the two populations be normally distributed, however because the Welch procedure is only an approximate procedure, it does not require this assumption.@
qu.1.2.choice.5=The pooled-variance t procedure is most appropriate when the observations between the two groups are dependent.@
qu.1.2.fixed=@

qu.2.topic=Population Standard Deviations Known@

qu.2.1.mode=Inline@
qu.2.1.name=Calculate test statistic, p-value, conclusion for one-sided hypothesis test (2)@
qu.2.1.comment=<p>a)&nbsp; The <em>z</em> test statistic is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>z</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><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate the test statistic 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><mi>z</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><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><msqrt><mrow><mfrac><mi>$var1</mi><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mi>$var2</mi><mrow><mi>$n2</mi></mrow></mfrac></mrow></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ZTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates we are performing a one-sided, upper tailed test.&nbsp; Therefore, the p-value is the area to the right of the test statistic, under a standard normal distribution.&nbsp; Because the test statistic is a large negative number, the area to the right of it&nbsp;will also quite large.&nbsp; Using computer&nbsp;software, this area is found to be approximately&nbsp;<em>p-value = $pvalue</em>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value is so large, there is no significant evidence against the null hypothesis, and therefore the null hypothesis is not rejected.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$n1=range(140, 170);
$n2=range(150, 180);
$xbar1=rand(31, 33, 3);
$xbar2=rand(35, 38, 3);
condition:lt($xbar1,$xbar2);
$var1=rand(25, 27, 3);
$var2=rand(21, 23, 3);
$PE=$xbar1-$xbar2;
$SE=sqrt(($var1/$n1) + ($var2/$n2));
$ZTest=$PE/$SE;
$pvalue=1-erf($ZTest);@
qu.2.1.uid=d8645545-7449-4a82-9f83-f1b8b47ca81f@
qu.2.1.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.1.weighting=1,1,1@
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qu.2.1.part.1.err=0.0010@
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qu.2.1.part.1.answer.num=$ZTest@
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qu.2.1.part.2.choice.5=-<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&infin;</mi></mrow></mstyle></math>@
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qu.2.1.part.3.choice.5=Due to the small p-value, we can be certain that the alternative hypothesis is correct.@
qu.2.1.part.3.fixed=@
qu.2.1.part.3.choice.4=There is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis, due to the small p-value.@
qu.2.1.part.3.question=null@
qu.2.1.part.3.choice.3=Because the p-value is large, there is insufficient evidence to show that the alternative hypothesis is correct.@
qu.2.1.part.3.choice.2=Due to the large p-value, we have no significant evidence against the null hypothesis.@
qu.2.1.part.3.choice.1=We&nbsp;have very strong evidence&nbsp;that the null hypothesis is correct, as the p-value is very large.@
qu.2.1.part.3.mode=Multiple Choice@
qu.2.1.part.3.display=vertical@
qu.2.1.part.3.answer=2@
qu.2.1.question=<p>Independent random samples of&nbsp; size&nbsp;<em>n<sub>1</sub>= $n1</em>&nbsp; and <em>n<sub>2</sub> = $n2</em> are drawn from two normally distributed populations with known variances of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var2</mi></mrow></mstyle></math>, respectively.&nbsp; The sample means are estimated to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><msub><mi>x</mi><mrow><mi></mi></mrow></msub></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi></mrow></mstyle></math>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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math>.&nbsp; 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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the test statistic?</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places.</p><p><span>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p>b)&nbsp; What is the approximate value of the p-value?</p><p>&nbsp;</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>c)&nbsp; At the 1% level of significance, what is the appropriate conclusion?</span></span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p>@

qu.2.2.mode=Inline@
qu.2.2.name=Calculate test statistic, rejection region, conclusion for two-sided hypothesis test@
qu.2.2.comment=<p>a)&nbsp; The <em>z</em> test statistic is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>z</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><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mfrac><mrow><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate the test statistic 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><mi>z</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><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><msqrt><mrow><mfrac><mi>$var1</mi><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mi>$var2</mi><mrow><mi>$n2</mi></mrow></mfrac></mrow></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ZTest</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Since the alternative hypothesis indicates we are conducting a two-sided hypothesis test, the rejection region is determined by a&nbsp;<em>z</em> value such that the areas above <em>z</em> and below <em>-z</em> are each equal to 0.025.&nbsp; Using computer software, or approximating with a standard normal table, this&nbsp;<em>z</em> value is equal to +/- 1.96.&nbsp; Therefore, the rejection region is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1.96</mn></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the <em>z</em>&nbsp; test statistic is greater than 1.96, there is sufficient evidence to reject the null hypothesis at the 5% level of significance.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$n1=range(30, 35);
$n2=range(30, 35);
condition:not(eq($n1,$n2));
$xbar1=rand(33, 35, 3);
$xbar2=rand(32, 34, 3);
condition:gt($xbar1,$xbar2);
$var1=rand(2, 4, 2);
$var2=rand(1.1, 3, 2);
$PE=$xbar1-$xbar2;
$SE=sqrt(($var1/$n1) + ($var2/$n2));
$ZTest=$PE/$SE;
condition:gt($ZTest,1.96);@
qu.2.2.uid=375c207b-d2f2-44c9-b4d7-4d8e4533b9cd@
qu.2.2.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.2.2.question=<p>Independent random samples of approximately the same size&nbsp;(<em>n<sub>1</sub>= $n1</em>&nbsp; and <em>n<sub>2</sub> = $n2,&nbsp;</em>respectively) are drawn from two normally distributed populations with known variances of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var2</mi></mrow></mstyle></math>, respectively.&nbsp; The sample means are calculated to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xbar1</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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math>.&nbsp; 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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the test statistic?</p><p>&nbsp;</p><p>Round your answer to at least 3 decimal places.</p><p><span>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>b)&nbsp; At the 5% level of significance, what is the appropriate rejection region for the test?</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>c)&nbsp; Is there sufficient evidence to reject the null hypothesis at the 5% level of significance?</span></span></span></p><p><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p>@

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qu.2.3.name=Calculate test statistic, p-value, conclusion for one-sided hypothesis test (1)@
qu.2.3.comment=<p>a)&nbsp; The <em>z</em> test statistic is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mfrac><mrow><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate the <em>z</em> test statistic 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><mi>z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><msqrt><mrow><mfrac><mi>$var1</mi><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mi>$var2</mi><mrow><mi>$n2</mi></mrow></mfrac></mrow></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ZTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are performing a one-sided, lower tailed test, which means that the p-value is the area under to the left of the test statistic, under&nbsp;a standard normal curve.&nbsp; Using computer software, or approximating with a standard normal table, we can find this area to be <em>p-value = $pvalue</em>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value is less than&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis in favour of the alternative hypothesis at the 5% level of significance.&nbsp; Therefore, there is sufficient evidence to conclude that the mean for Population 1 is less than the mean for Population 2.</p>@
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qu.2.3.algorithm=$n1=range(14, 17);
$n2=range(15, 18);
$xbar1=rand(1.1, 1.3, 3);
$xbar2=rand(1.5, 1.8, 3);
condition:lt($xbar1,$xbar2);
$var1=rand(0.2, 0.4, 2);
$var2=rand(0.1, 0.3, 2);
$PE=$xbar1-$xbar2;
$SE=sqrt(($var1/$n1) + ($var2/$n2));
$ZTest=$PE/$SE;
$pvalue=erf($ZTest);
condition:lt($pvalue,0.05);
condition:gt($pvalue,0.001);@
qu.2.3.uid=5386eb91-52b5-4fcd-8dce-dba57a6f3daf@
qu.2.3.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.2.3.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore no significant&nbsp;evidence that the means of the two populations are not equal.@
qu.2.3.part.3.choice.1=There is sufficient evidence to reject the null hypothesis, and&nbsp;therefore evidence that&nbsp;the mean of Population 1 is less than the mean of Population 2.@
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qu.2.3.question=<p>Independent random samples of&nbsp; size&nbsp;<em>n<sub>1</sub>= $n1</em>&nbsp; and <em>n<sub>2</sub> = $n2</em> are drawn from two normally distributed populations with known variances of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var2</mi></mrow></mstyle></math>, respectively.&nbsp; The sample means are calculated to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi></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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math>.&nbsp; 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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the test statistic?</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places.</p><p><span>&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>b)&nbsp; What is the approximate value of the p-value?</span></span></p><p>&nbsp;</p><p><span><span>Round your response to at least&nbsp;3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p><span><span><span>c)&nbsp; At the 5% level of significance, what is the appropriate conclusion?</span></span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p>@

qu.2.4.mode=Inline@
qu.2.4.name=Calculate point estimate, standard error, margin of error for 90% confidence interval@
qu.2.4.comment=<p>a)&nbsp; The point estimate 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>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>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><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$PE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The standard error of the difference in sample 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></mi></mrow><mrow><mi></mi></mrow><mrow><msub><mi>&sigma;</mi><mrow><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow><mrow><mi></mi></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><mrow><msub><mi>&sigma;</mi><mrow><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><msubsup><mi>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><msub><mi>&sigma;</mi><mrow><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><mrow><mi>$var1</mi></mrow><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mrow><mi>$var2</mi></mrow><mrow><mi>$n2</mi></mrow></mfrac></mrow></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; For a 90% confidence interval for the difference in population 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><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1.645</mn></mrow></mstyle></math>, and the margin of error for the confidence interval 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='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><msub><mi>&sigma;</mi><mrow><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values results in a margin of 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>ME</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>1.645</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME</mi></mrow></mstyle></math>.</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$n1=range(60, 65);
$n2=range(45, 50);
$xbar1=rand(10, 12, 3);
$xbar2=rand(9, 11, 3);
condition:gt($xbar1,$xbar2);
$var1=rand(20, 24, 3);
$var2=rand(31, 35, 3);
$PE=$xbar1-$xbar2;
$SE=sqrt(($var1/$n1) + ($var2/$n2));
$ME=1.645*$SE;@
qu.2.4.uid=08bf40bf-6109-4973-ae9f-88954eceb943@
qu.2.4.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.2.4.weighting=1,1,1@
qu.2.4.numbering=alpha@
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qu.2.4.part.1.grading=exact_value@
qu.2.4.part.1.negStyle=both@
qu.2.4.part.1.answer.num=$PE@
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qu.2.4.part.2.numStyle=   @
qu.2.4.part.2.editing=useHTML@
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qu.2.4.part.2.err=0.0010@
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qu.2.4.part.2.negStyle=both@
qu.2.4.part.2.answer.num=$SE@
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qu.2.4.part.3.editing=useHTML@
qu.2.4.part.3.showUnits=false@
qu.2.4.part.3.err=0.0010@
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qu.2.4.part.3.grading=toler_abs@
qu.2.4.part.3.negStyle=both@
qu.2.4.part.3.answer.num=$ME@
qu.2.4.question=<p>Independent random samples are drawn from two normally distributed populations, with known population variances of&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>&sigma;</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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>&sigma;</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var2</mi></mrow></mstyle></math>.&nbsp; The following summary statistics are obtained:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Sample 1</strong></p>            </td>            <td>            <p align="center"><strong>Sample 2</strong></p>            </td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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><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>$xbar2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi></mrow></mstyle></math></td>            <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>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the point estimate 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>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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></mi></mrow><mrow><msub><mi>&sigma;</mi><mrow><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mrow><mover><mi>X</mi><mi>&macr;</mi></mover></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></mstyle></math>?</span></p><p>&nbsp;</p><p><span>Round your response to at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; What is the margin of error 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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>?</span></span></p><p>&nbsp;</p><p><span><span>Round your response to at least&nbsp;3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.3.topic=Pooled-Variance t Procedure@

qu.3.1.mode=Inline@
qu.3.1.name=Determine test statistic, range of p-value, conclusion for one-sided test (1)@
qu.3.1.comment=<p>a)&nbsp; The formula for the <em>t</em> test statistic is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></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>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</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></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>1</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>2</mn></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; The pooled standard deviation is obtained by first calculating the pooled variance, 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><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><msub><mi>n</mi><mrow><mn>1</mn></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><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>n</mi><mrow><mn>2</mn></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><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><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></mrow></mfenced><mrow><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp2</mi></mrow></mstyle></math>, and therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mi>p</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$sp2</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp</mi></mrow></mstyle></math>.&nbsp; Using this value, along with the other given information, the test statistic is calculated to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><mi>$sp</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mi>$n2</mi></mrow></mfrac></mrow></mrow></msqrt></mrow></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are conducting a one-sided, lower tailed test.&nbsp; Therefore, the p-value is the area to the left of the test statistic, under&nbsp;a <em>t</em> distribution with&nbsp;<em>$n1 + $n2 -&nbsp;2 = $df</em> degrees of freedom.&nbsp; Using computer software, we can find this area to be <em>p-value&nbsp;= $pvalue</em>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value is very small, less than both&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.01</mn></mrow></mstyle></math>, the null hypothesis is rejected at both the 5% and 1% levels of significance.</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$n1=range(15,20);
$n2=range(15,20);
$df=$n1+$n2-2;
$xbar1=rand(20,24,4);
$xbar2=rand(24,28,4);
$var1=rand(2,3,3);
$var2=rand(3,4,3);
$sp2=(($n1-1)*$var1 + ($n2-1)*$var2)/($n1+$n2-2);
$sp=sqrt($sp2);
$tTest=($xbar1-$xbar2)/($sp*sqrt(1/$n1 + 1/$n2));
$pvalue=studentst($df,$tTest);
condition:lt($pvalue,0.005);@
qu.3.1.uid=c68e1b56-f638-41ce-9362-dbcf920a9e20@
qu.3.1.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
qu.3.1.weighting=1,1,1@
qu.3.1.numbering=alpha@
qu.3.1.part.1.name=sro_id_1@
qu.3.1.part.1.answer.units=@
qu.3.1.part.1.numStyle=   @
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.showUnits=false@
qu.3.1.part.1.err=0.0010@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.mode=Numeric@
qu.3.1.part.1.grading=toler_abs@
qu.3.1.part.1.negStyle=both@
qu.3.1.part.1.answer.num=$tTest@
qu.3.1.part.2.name=sro_id_2@
qu.3.1.part.2.editing=useHTML@
qu.3.1.part.2.choice.5=p-value < 0.01@
qu.3.1.part.2.fixed=@
qu.3.1.part.2.choice.4=0.01 < p-value < 0.025@
qu.3.1.part.2.question=null@
qu.3.1.part.2.choice.3=0.025 < p-value < 0.05@
qu.3.1.part.2.choice.2=0.05 < p-value < 0.10@
qu.3.1.part.2.choice.1=p-value > 0.10@
qu.3.1.part.2.mode=Non Permuting Multiple Choice@
qu.3.1.part.2.display=vertical@
qu.3.1.part.2.answer=5@
qu.3.1.part.3.name=sro_id_3@
qu.3.1.part.3.editing=useHTML@
qu.3.1.part.3.fixed=@
qu.3.1.part.3.choice.4=There is insufficient evidence to reject the null hypothesis at both the 5% and 1% level of significance.@
qu.3.1.part.3.question=null@
qu.3.1.part.3.choice.3=There is sufficient evidence to reject the null hypothesis at the 1% level of significance, but not at the 5% level of significance.@
qu.3.1.part.3.choice.2=There is sufficient evidence to reject the null hypothesis at the 5% level of significance, but not at the 1% level of significance.@
qu.3.1.part.3.choice.1=There is sufficient evidence to reject the null hypothesis at both the 5% and 1% level of significance.@
qu.3.1.part.3.mode=Multiple Choice@
qu.3.1.part.3.display=vertical@
qu.3.1.part.3.answer=1@
qu.3.1.question=<p>Consider the following summary statistics, calculated from two independent&nbsp;random samples taken from normally distributed&nbsp;populations.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Sample 1</strong></p>            </td>            <td>            <p align="center"><strong>Sample 2</strong></p>            </td>        </tr>        <tr>            <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><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>$xbar2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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><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>$var2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi></mrow></mstyle></math></td>            <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>    </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><msub><mi mathvariant='normal'>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>.</p><p>&nbsp;</p><p><span>a)&nbsp; Calculate the test statistic for the pooled-variance <em>t</em> procedure.</span></p><p>&nbsp;</p><p><span>Round your response to at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>b)&nbsp; Determine the range in which the p-value falls:</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>c)&nbsp; What is the most appropriate conclusion that can be made?</span></span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p>@

qu.3.2.mode=Inline@
qu.3.2.name=Determine hypothesis, test statistic, range of p-value for two-sided test@
qu.3.2.comment=<p>a)&nbsp; Since we are only interested in determining if there is a difference between the population means, the implication is that we are conducting a two-sided hypothesis test.&nbsp; Therefore, the appropriate null and alternative hypotheses 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>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><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></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>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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&ne;</mo></mrow><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The <em>t</em> test statistic is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mrow><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover></mrow><msub><mi></mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mrow><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced></mrow></mfrac></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>1</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>2</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>1</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>2</mn></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; To calculate the pooled standard deviation, we can first calculate the pooled variance: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><msub><mi>n</mi><mrow><mn>1</mn></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><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>n</mi><mrow><mn>2</mn></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><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><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='0.0em' 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>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var2</mi></mrow><mrow><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp2</mi></mrow></mstyle></math>, and therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mi>p</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$sp2</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp</mi></mrow></mstyle></math>.&nbsp; Substituting these values into the equation for the test statistic, along with the other given information, results in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><mi>$sp</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mi>$n2</mi></mrow></mfrac></mrow></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the alternative hypothesis indicates that we are conducting a two-sided hypothesis test, the p-value is determined as&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tTest</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>,&nbsp;where <em>t</em> follows a <em>t</em> distribution with <em>$n1 + $n2 -&nbsp;2 = $df</em> degrees of freedom.&nbsp; Using computer software, we can find the area in the tail&nbsp;to be exactly&nbsp;<em>$Tail, </em>and therefore the p-value is <em>2 X $Tail = $pvalue</em>.</p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$n1=range(7,10);
$n2=range(9,12);
$df=$n1+$n2-2;
$xbar1=rand(5,6,4);
$xbar2=rand(7,8,4);
$var1=rand(2,3,4);
$var2=rand(0.5,1.5,4);
$sp2=(($n1-1)*$var1 + ($n2-1)*$var2)/($n1+$n2-2);
$sp=sqrt($sp2);
$tTest=($xbar1-$xbar2)/($sp*sqrt(1/$n1 + 1/$n2));
$Tail=studentst($df,$tTest);
$pvalue=$Tail*2;
condition:lt($pvalue,0.025);@
qu.3.2.uid=acb6bda6-cbf4-456d-ad66-e2529c982fc3@
qu.3.2.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
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qu.3.2.part.3.choice.5=p-value < 0.025@
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qu.3.2.part.3.choice.4=0.025 < p-value < 0.05@
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qu.3.2.question=<p>Consider the following summary statistics, calculated from two independent random samples taken from normally distributed&nbsp;populations.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Sample 1</strong></p>            </td>            <td>            <p align="center"><strong>Sample 2</strong></p>            </td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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><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>$xbar2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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><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>$var2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi></mrow></mstyle></math></td>            <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>    </tbody></table></p><p>&nbsp;</p><p>You are interested in determining if there is sufficient evidence of a difference between the population means.</p><p>&nbsp;</p><p>a)&nbsp; What are the appropriate null and alternative hypotheses?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; Calculate the test statistic for the pooled-variance <em>t</em> procedure.</span></p><p>&nbsp;</p><p><span>Round your response to at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Determine the range in which the p-value falls:</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.3.3.mode=Inline@
qu.3.3.name=Calculate pooled variance, degrees of freedom@
qu.3.3.comment=<p>a)&nbsp; The pooled variance, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>, is calculated 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><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><msub><mi>n</mi><mrow><mn>1</mn></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><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>n</mi><mrow><mn>2</mn></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><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>$n1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$n2</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><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp2</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The degrees of freedom for the pooled-variance <em>t</em> procedure are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mstyle></math>.&nbsp; Therefore, in this case the degrees of freedom are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$n1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$df</mi></mrow></mstyle></math>.</p>@
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$n2=range(10,15);
$s1=rand(4.5, 5.5, 2);
$s2=rand(4, 5, 2);
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qu.3.3.uid=fb920c62-8cc4-4656-910c-864f2f6eb597@
qu.3.3.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.3.3.question=<p>Consider two independent&nbsp;random samples of sizes&nbsp;<em>n<sub>1 </sub>=&nbsp;$n1</em> and <em>n<sub>2</sub> =&nbsp;$n2</em>, taken from normally distributed populations, with sample standard deviations of <em>s<sub>1</sub> = $s1</em> and <em>s<sub>2</sub> = $s2, </em>respectively.</p><p>&nbsp;</p><p>a)&nbsp; What is s<sub>p</sub><sup>2</sup>, the estimate of the pooled variance?</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What are the appropriate degrees of freedom for the <em>t</em>&nbsp; value?</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p>@

qu.3.4.mode=Inline@
qu.3.4.name=Calculate point estimates, 90% confidence interval@
qu.3.4.comment=<p>a)&nbsp; To calculate a point estimate 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>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>, we first need to calculate the sample mean for each of the two samples: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><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>6</mn></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>5</mn></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn>5</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math>.&nbsp; Therefore, a point estimate for the difference between populations means 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><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$xbar2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$PEMean</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The estimate for the pooled standard deviation can be found by taking the square root of the pooled variance, which 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><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><msub><mi>n</mi><mrow><mn>1</mn></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><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>n</mi><mrow><mn>2</mn></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><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac></mrow></mstyle></math>.&nbsp; The sample variances are 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><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><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>6</mn></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><mi>$xbar1</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>5</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var1</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>5</mn></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var2</mi></mrow></mstyle></math>.&nbsp; Substituting these values into the equation for pooled variance, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mn>6</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var1</mi><mo lspace='0.0em' 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><mn>5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$var2</mi></mrow><mrow><mn>6</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>5</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$PooledVar</mi></mrow></mstyle></math>.&nbsp; Therefore, the estimate for the pooled standard deviation 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>s</mi><mrow><mi>p</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$PooledVar</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$PooledSD</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; 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><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>100</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>%</mi></mrow></mstyle></math>confidence interval for the difference in population means 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><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></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>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</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></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>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</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></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>1</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>2</mn></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; The degrees of freedom for the <em>t</em> distribution 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>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>6</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>9</mn></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>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>$tAlpha2</mi></mrow></mstyle></math>.&nbsp; Using the values found in parts (a) and (b), we can calculate a 90% confidence interval to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>$PEMean</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&pm;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$PooledSD</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mn>5</mn></mrow></mfrac></mrow></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$LL</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$UL</mi></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.3.4.editing=useHTML@
qu.3.4.solution=@
qu.3.4.algorithm=$S1=[rand(6,14,3),rand(6,14,3),rand(6,14,3),rand(6,14,3),rand(6,14,3),rand(6,14,3)];
$S2=[rand(5,18,3),rand(5,18,3),rand(5,18,3),rand(5,18,3),rand(5,18,3)];
$Data=maple("
Avg1:=Statistics[Mean]($S1):
Avg2:=Statistics[Mean]($S2):
Var1:=Statistics[Variance]($S1):
Var2:=Statistics[Variance]($S2):
Avg1, Avg2, Var1, Var2
");
$xbar1=switch(0, $Data);
$xbar2=switch(1, $Data);
$var1=switch(2, $Data);
$var2=switch(3, $Data);
$PEMean=$xbar1-$xbar2;
$PooledVar=(5*$var1 + 4*$var2)/9;
$PooledSD=sqrt($PooledVar);
$tAlpha2=invstudentst(9, 0.95);
$ME=$tAlpha2*$PooledSD*sqrt(1/6 + 1/5);
$UL=$PEMean + $ME;
$LL=$PEMean - $ME;@
qu.3.4.uid=0457202b-6ea5-450e-bfd7-860e25f1fadb@
qu.3.4.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
qu.3.4.weighting=1,1,1@
qu.3.4.numbering=alpha@
qu.3.4.part.1.name=sro_id_1@
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qu.3.4.part.1.numStyle=   @
qu.3.4.part.1.editing=useHTML@
qu.3.4.part.1.showUnits=false@
qu.3.4.part.1.err=0.0010@
qu.3.4.part.1.question=(Unset)@
qu.3.4.part.1.mode=Numeric@
qu.3.4.part.1.grading=toler_abs@
qu.3.4.part.1.negStyle=both@
qu.3.4.part.1.answer.num=$PEMean@
qu.3.4.part.2.name=sro_id_2@
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qu.3.4.part.2.numStyle=   @
qu.3.4.part.2.editing=useHTML@
qu.3.4.part.2.showUnits=false@
qu.3.4.part.2.err=0.01@
qu.3.4.part.2.question=(Unset)@
qu.3.4.part.2.mode=Numeric@
qu.3.4.part.2.grading=toler_abs@
qu.3.4.part.2.negStyle=both@
qu.3.4.part.2.answer.num=$PooledSD@
qu.3.4.part.3.editing=useHTML@
qu.3.4.part.3.question=(Unset)@
qu.3.4.part.3.name=sro_id_3@
qu.3.4.part.3.answer=($LL?0.01, $UL?0.01)@
qu.3.4.part.3.mode=Ntuple@
qu.3.4.question=<p>Consider the following data taken from two independent, normally distributed populations:</p><p>&nbsp;</p><p align="center">&nbsp;<strong>Sample 1: </strong>$S1</p><p align="center">&nbsp;</p><p align="center"><strong>Sample 2: </strong>$S2</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Calculate a point estimate 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>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></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 pooled standard deviation, <em>s<sub>p</sub>.</em></span></p><p>&nbsp;</p><p><span>Round your response to at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Calculate 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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>.</span></span></p><p>&nbsp;</p><p><span><span>Enter your response in the format: <strong>(lower limit, upper limit)</strong>.</span></span></p><p>&nbsp;</p><p><span><span>Round your responses to at least&nbsp;3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.3.5.mode=Inline@
qu.3.5.name=Calculate pooled variance, margin of error for 95% confidence interval@
qu.3.5.comment=<p>a)&nbsp; The formula for the pooled variance 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><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><msub><mi>n</mi><mrow><mn>1</mn></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><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>n</mi><mrow><mn>2</mn></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><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate the pooled variance 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><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><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The appropriate degrees of freedom for the <em>t</em> distribution to be used is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$df</mi></mrow></mstyle></math>.&nbsp; Therefore, for a 95% confidence interval <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The margin of error for a 95% confidence interval for the difference between two means is given as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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>1</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>2</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>1</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>2</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>1</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>2</mn></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; Substituting in the values from parts (a) and (b), we can calculate the margin of error 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><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi>$sp2</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mi>$n2</mi></mrow></mfrac></mrow></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME</mi></mrow></mstyle></math>.</p>@
qu.3.5.editing=useHTML@
qu.3.5.solution=@
qu.3.5.algorithm=$n1=range(8,13);
$n2=range(8,13);
$s1=rand(2.5, 4.5, 2);
$s2=rand(2, 4, 2);
$df=$n1+$n2-2;
$sp2=(($n1-1)*$s1^2 + ($n2-1)*$s2^2)/$df;
$sp=sqrt($sp2);
$tAlpha2=invstudentst($df, 0.975);
$ME=$tAlpha2*$sp*sqrt(1/$n1 + 1/$n2);@
qu.3.5.uid=1e192e7f-02a1-4bd3-b0e4-ac53c448fb0f@
qu.3.5.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.3.5.weighting=1,1,1@
qu.3.5.numbering=alpha@
qu.3.5.part.1.name=sro_id_1@
qu.3.5.part.1.answer.units=@
qu.3.5.part.1.numStyle=   @
qu.3.5.part.1.editing=useHTML@
qu.3.5.part.1.showUnits=false@
qu.3.5.part.1.err=0.0010@
qu.3.5.part.1.question=(Unset)@
qu.3.5.part.1.mode=Numeric@
qu.3.5.part.1.grading=toler_abs@
qu.3.5.part.1.negStyle=both@
qu.3.5.part.1.answer.num=$sp2@
qu.3.5.part.2.name=sro_id_2@
qu.3.5.part.2.answer.units=@
qu.3.5.part.2.numStyle=   @
qu.3.5.part.2.editing=useHTML@
qu.3.5.part.2.showUnits=false@
qu.3.5.part.2.err=0.0010@
qu.3.5.part.2.question=(Unset)@
qu.3.5.part.2.mode=Numeric@
qu.3.5.part.2.grading=toler_abs@
qu.3.5.part.2.negStyle=both@
qu.3.5.part.2.answer.num=$tAlpha2@
qu.3.5.part.3.name=sro_id_3@
qu.3.5.part.3.answer.units=@
qu.3.5.part.3.numStyle=   @
qu.3.5.part.3.editing=useHTML@
qu.3.5.part.3.showUnits=false@
qu.3.5.part.3.err=0.01@
qu.3.5.part.3.question=(Unset)@
qu.3.5.part.3.mode=Numeric@
qu.3.5.part.3.grading=toler_abs@
qu.3.5.part.3.negStyle=both@
qu.3.5.part.3.answer.num=$ME@
qu.3.5.question=<p>Consider two independent&nbsp;random samples of sizes&nbsp;<em>n<sub>1 </sub>=&nbsp;$n1</em> and <em>n<sub>2</sub> =&nbsp;$n2</em>, taken from normally distributed&nbsp;populations with sample standard deviations of <em>s<sub>1</sub> = $s1</em> and <em>s<sub>2</sub> = $s2, </em>respectively.</p><p>&nbsp;</p><p>a)&nbsp; What is s<sub>p</sub><sup>2</sup>, the estimate of the pooled variance?</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the 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>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>&nbsp;value 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>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>?</span></p><p>&nbsp;</p><p>Round your response to at least&nbsp;3 decimal places.</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Using your responses from parts (a) and (b), what is the value for the margin of error?</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>@

qu.3.6.mode=Inline@
qu.3.6.name=Determine test statistic, range of p-value, conclusion for one-sided test (2)@
qu.3.6.comment=<p>a)&nbsp; The estimate of the pooled standard deviation can be found by first calculating the pooled variance, 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><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><msub><mi>n</mi><mrow><mn>1</mn></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><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>n</mi><mrow><mn>2</mn></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><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><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></mrow><mrow><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp2</mi></mrow></mstyle></math>, and therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mi>p</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$sp2</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The <em>t</em> test statistic is calculated 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><mi>t</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><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced></mrow></mfrac></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>1</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>2</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>1</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>2</mn></mrow></msub></mrow></mfrac></mrow></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate the test statistic 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><mi>t</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><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><mi>$sp</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mfrac><mn>1</mn><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mn>1</mn><mrow><mi>$n2</mi></mrow></mfrac></mrow></mrow></msqrt></mrow></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The p-value is the area to the right of the test statistic,&nbsp;under&nbsp;a <em>t</em> distribution with&nbsp;<em>$n1 + $n2 - 2 = $df</em> degrees of freedom,&nbsp;as the alternative hypothesis indicates that we are performing a one-sided, upper tailed test.&nbsp; Using computer software, we can find this area to be exactly <em>p-value = $pvalue</em>.&nbsp; Since this p-value is very large, there is insufficient evidence to reject the null hypothesis, and therefore no significant&nbsp;evidence that the population means are not equal to each other.</p>@
qu.3.6.editing=useHTML@
qu.3.6.solution=@
qu.3.6.algorithm=$n1=range(15,20);
$n2=range(15,20);
$df=$n1+$n2-2;
$xbar1=rand(120,124,4);
$xbar2=rand(122,126,4);
$var1=rand(6,7,3);
$var2=rand(7,8,3);
$sp2=(($n1-1)*$var1 + ($n2-1)*$var2)/($n1+$n2-2);
$sp=sqrt($sp2);
$tTest=($xbar1-$xbar2)/($sp*sqrt(1/$n1 + 1/$n2));
$pvalue=1-studentst($df,$tTest);@
qu.3.6.uid=4bfbc8d4-804d-47c4-85f3-8f45aea910c5@
qu.3.6.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
qu.3.6.weighting=1,1,1@
qu.3.6.numbering=alpha@
qu.3.6.part.1.name=sro_id_1@
qu.3.6.part.1.answer.units=@
qu.3.6.part.1.numStyle=   @
qu.3.6.part.1.editing=useHTML@
qu.3.6.part.1.showUnits=false@
qu.3.6.part.1.err=0.0010@
qu.3.6.part.1.question=(Unset)@
qu.3.6.part.1.mode=Numeric@
qu.3.6.part.1.grading=toler_abs@
qu.3.6.part.1.negStyle=both@
qu.3.6.part.1.answer.num=$sp@
qu.3.6.part.2.name=sro_id_2@
qu.3.6.part.2.answer.units=@
qu.3.6.part.2.numStyle=   @
qu.3.6.part.2.editing=useHTML@
qu.3.6.part.2.showUnits=false@
qu.3.6.part.2.err=0.0010@
qu.3.6.part.2.question=(Unset)@
qu.3.6.part.2.mode=Numeric@
qu.3.6.part.2.grading=toler_abs@
qu.3.6.part.2.negStyle=both@
qu.3.6.part.2.answer.num=$tTest@
qu.3.6.part.3.comment.2=@
qu.3.6.part.3.name=sro_id_3@
qu.3.6.part.3.comment.1=@
qu.3.6.part.3.editing=useHTML@
qu.3.6.part.3.fixed=@
qu.3.6.part.3.question=null@
qu.3.6.part.3.choice.2=At the 10% level of significance, there is insufficient evidence to reject the null hypothesis, and therefore no significant&nbsp;evidence that the population means are not equal to each other.@
qu.3.6.part.3.choice.1=At the 10% level of significance, there is sufficient evidence to reject the null hypothesis and therefore evidence that the mean of Population 1 is greater than the mean of Population 2.@
qu.3.6.part.3.mode=Non Permuting Multiple Choice@
qu.3.6.part.3.display=vertical@
qu.3.6.part.3.answer=2@
qu.3.6.question=<p>Consider the following summary statistics, calculated from two independent&nbsp;random samples taken from normally distributed&nbsp;populations.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Sample 1</strong></p>            </td>            <td>            <p align="center"><strong>Sample 2</strong></p>            </td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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><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>$xbar2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$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><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>$var2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi></mrow></mstyle></math></td>            <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>    </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><msub><mi mathvariant='normal'>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, at the 10% level of significance.</p><p>&nbsp;</p><p>a)&nbsp; Calculate the pooled standard deviation, <em>s<sub>p</sub></em>.</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; Calculate the test statistic for the pooled-variance <em>t</em> procedure.</span></p><p>&nbsp;</p><p><span>Round your response to at least&nbsp;3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp;&nbsp;What is the appropriate conclusion that can be made?</span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></span></p>@

qu.4.topic=Welch Approximate t Procedure@

qu.4.1.mode=Inline@
qu.4.1.name=Calculate standard error, approximate degrees of freedom@
qu.4.1.comment=<p>a)&nbsp; 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>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>, using the Welch Approximate <em>t</em> procedure, is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate&nbsp; 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><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n2</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SEWelch</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The degrees of freedom using the Welch-Satterthwaite approximation&nbsp;is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>df</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>.&nbsp; When the appropriate values are substituted in, the degrees of freedom are approximated as&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>df</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n2</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n1</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n2</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$DFWS</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The conservative degrees of freedom for the Welch Approximate <em>t</em> procecure is the smaller 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><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>&nbsp;and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>, which in this case is $DFConservative.&nbsp;</p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$n1=range(10,15);
$n2=range(10,15);
$s1=rand(4,6,2);
$var1=$s1^2;
$s2=rand(8,10,2);
$var2=$s2^2;
$SEWelch=sqrt(($var1/$n1)+($var2/$n2));
$DFConservative=min($n1-1, $n2-1);
$DFWS=(($var1/$n1 + $var2/$n2)^2)/(((1/($n1-1))*($var1/$n1)^2) + ((1/($n2-1))*($var2/$n2)^2));@
qu.4.1.uid=c807dfd9-8167-49f9-ac4d-c5827f23f1b4@
qu.4.1.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.4.1.part.1.name=sro_id_1@
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qu.4.1.part.1.showUnits=false@
qu.4.1.part.1.err=0.0010@
qu.4.1.part.1.question=(Unset)@
qu.4.1.part.1.mode=Numeric@
qu.4.1.part.1.grading=toler_abs@
qu.4.1.part.1.negStyle=both@
qu.4.1.part.1.answer.num=$SEWelch@
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qu.4.1.part.2.numStyle=   @
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qu.4.1.part.2.grading=toler_abs@
qu.4.1.part.2.negStyle=both@
qu.4.1.part.2.answer.num=$DFWS@
qu.4.1.part.3.name=sro_id_3@
qu.4.1.part.3.answer.units=@
qu.4.1.part.3.numStyle=   @
qu.4.1.part.3.editing=useHTML@
qu.4.1.part.3.showUnits=false@
qu.4.1.part.3.question=(Unset)@
qu.4.1.part.3.mode=Numeric@
qu.4.1.part.3.grading=exact_value@
qu.4.1.part.3.negStyle=both@
qu.4.1.part.3.answer.num=$DFConservative@
qu.4.1.question=<p>Consider two independent&nbsp;random samples of sizes <em>n<sub>1</sub> = $n1 </em>and <em>n<sub>2</sub> = $n2</em>, taken from two normally distributed populations.&nbsp; The sample standard deviations were calculated to be <em>s<sub>1</sub>= $s1</em>&nbsp; and <em>s<sub>2</sub>&nbsp; = $s2</em>, respectively.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Calculate the standard error of the difference in sample 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><mn>1</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>2</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>, using the Welch Approximate t procedure.</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; Estimate the corresponding degrees of freedom, using the Welch-Satterthwaite approximation.</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; If instead of using the Welch-Satterthwaite approximation of the degrees of freedom, the conservative method for estimating the degrees of freedom is used, what would be the appropriate degrees of freedom?</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.4.2.mode=Inline@
qu.4.2.name=Calculate test statistic, p-value for two-sided hypothesis test@
qu.4.2.comment=<p>a)&nbsp; Using the Welch Approximate <em>t</em> procedure, the <em>t</em> test statistic is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we calculate the test statistic 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><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><msqrt><mrow><mfrac><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n2</mi></mrow></mfrac></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The conservative estimate of the degrees of freedom is the lesser of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></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>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>, which in this case is $DFConservative.&nbsp; Since the alternative hypothesis indicates that we are performing a two-sided hypothesis test, the p-value is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tTest</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>.&nbsp; Using computer software, we can find the area in the tail of the distribution to be <em>$Tail</em>, and therefore the p-value is found to be <em>2 X $Tail = $pvalue</em>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Yes, the pooled-variance <em>t</em> procedure would have also worked reasonably well in this case, since the sample variances <strong>and </strong>the sample sizes are not too different from each other.</p>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$n1=range(10,15);
$n2=range(10,15);
$s1=rand(34,36,3);
$var1=$s1^2;
$s2=rand(38,40,3);
$var2=$s2^2;
$xbar1=rand(-10,-5,3);
$xbar2=rand(-5,-0.01,3);
$PE=$xbar1-$xbar2;
$SEWelch=sqrt(($var1/$n1)+($var2/$n2));
$DFConservative=min($n1-1, $n2-1);
$tTest=$PE/$SEWelch;
$Tail=studentst($DFConservative, $tTest);
$pvalue=$Tail*2;
condition:gt($pvalue,0.50);@
qu.4.2.uid=38c2d551-41a0-4710-9ae0-5b8ff95f1aef@
qu.4.2.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.4.2.weighting=1,1,1@
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qu.4.2.part.1.editing=useHTML@
qu.4.2.part.1.showUnits=false@
qu.4.2.part.1.err=0.01@
qu.4.2.part.1.question=(Unset)@
qu.4.2.part.1.mode=Numeric@
qu.4.2.part.1.grading=toler_abs@
qu.4.2.part.1.negStyle=both@
qu.4.2.part.1.answer.num=$tTest@
qu.4.2.part.2.comment.3=@
qu.4.2.part.2.comment.2=@
qu.4.2.part.2.name=sro_id_2@
qu.4.2.part.2.comment.1=@
qu.4.2.part.2.editing=useHTML@
qu.4.2.part.2.choice.5=p-value < 0.01@
qu.4.2.part.2.fixed=@
qu.4.2.part.2.choice.4=0.01 < p-value < 0.025@
qu.4.2.part.2.question=null@
qu.4.2.part.2.choice.3=0.025 < p-value < 0.05@
qu.4.2.part.2.choice.2=0.05 < p-value < 0.10@
qu.4.2.part.2.choice.1=p-value > 0.10@
qu.4.2.part.2.mode=Non Permuting Multiple Choice@
qu.4.2.part.2.display=vertical@
qu.4.2.part.2.comment.5=@
qu.4.2.part.2.comment.4=@
qu.4.2.part.2.answer=1@
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qu.4.2.part.3.choice.4=No.&nbsp; The pooled-variance t procedure only performs well when the population variances are the same.@
qu.4.2.part.3.question=null@
qu.4.2.part.3.choice.3=Yes.&nbsp; Because the sample variances are not that different, it is likely that the population variances are the same, and the pooled-variance t procedure could be used.@
qu.4.2.part.3.choice.2=No.&nbsp; Because the sample variances are different, the pooled-variance t procedure should not be used.@
qu.4.2.part.3.choice.1=Yes.&nbsp; Because the sample variances are not too different, and the sample sizes are not too different, the pooled-variance t procedure would still perform relatively well.@
qu.4.2.part.3.mode=Multiple Choice@
qu.4.2.part.3.display=vertical@
qu.4.2.part.3.answer=1@
qu.4.2.question=<p>Consider two independent&nbsp;random samples of sizes <em>n<sub>1</sub> = $n1 </em>and <em>n<sub>2</sub> = $n2</em>, taken from two normally distributed populations.&nbsp; The sample standard deviations&nbsp;are calculated to be <em>s<sub>1</sub>= $s1</em>&nbsp; and <em>s<sub>2</sub>&nbsp; = $s2</em>, and the sample means&nbsp;are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi></mrow></mstyle></math>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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math>,&nbsp;respectively.&nbsp; Using this information, 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 mathvariant='normal'>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>against the two-sided alternative <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&ne;</mo></mrow><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>, using the Welch Approximate <em>t</em> Procedure.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Calculate&nbsp;the value for the <em>t</em> test statistic.</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp;&nbsp;Using&nbsp;a conservative estimate of the degrees of freedom (i.e. the lesser of <em>n<sub>1</sub> - 1</em>&nbsp;&nbsp;and <em>n<sub>2</sub> - 1</em>), the p-value is within which one of the following ranges?</span></p><p>&nbsp;</p><p><span>&nbsp;</span><2><span>&nbsp;</span>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>c)&nbsp; Would it have also been reasonable, in this situation, to have used the pooled-variance <em>t</em> procedure, rather than the Welch approximate <em>t</em> procedure?</p><p>&nbsp;</p><p>Pick one of the following that offers the best explanation.</p><p>&nbsp;</p><p><span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.4.3.mode=Inline@
qu.4.3.name=Calculate test statistic, conclusion for one-sided hypothesis test (2)@
qu.4.3.comment=<p>a)&nbsp; Using the Welch Approximate <em>t</em> procedure, the <em>t </em>test statistic is given by the equation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate the test statistic 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><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><msqrt><mrow><mfrac><mrow><mi>$var1</mi></mrow><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>$var2</mi></mrow><mrow><mi>$n2</mi></mrow></mfrac></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The conservative estimate of the degrees of freedom is the lesser of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></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>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>, which in this case is $DFConservative.&nbsp; The alternative hypothesis indicates that we are performing a one-sided, upper tailed test, which means that the p-value is determined as the area to the right of the test statistic, under a <em>t</em> distribution with $DFConservative degrees of freedom.&nbsp; Using computer software, we can find this area to be <em>p-value = $pvalue.</em></p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis at the 10% level of significance.&nbsp;</p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$n1=range(20,25);
$n2=range(8,13);
$s1=rand(1.1,3,3);
$var1=$s1^2;
$s2=rand(5,7,3);
$var2=$s2^2;
$xbar1=rand(2,6,3);
$xbar2=rand(3,7,3);
$PE=$xbar1-$xbar2;
$SEWelch=sqrt(($var1/$n1)+($var2/$n2));
$DFConservative=min($n1-1, $n2-1);
$tTest=$PE/$SEWelch;
$pvalue=1-studentst($DFConservative, $tTest);
condition:gt($pvalue,0.10);@
qu.4.3.uid=56e05d5c-8d5f-4012-94d4-170e07f638f3@
qu.4.3.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.4.3.part.2.choice.5=p-value < 0.01@
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qu.4.3.part.2.choice.4=0.01 < p-value < 0.025@
qu.4.3.part.2.question=null@
qu.4.3.part.2.choice.3=0.025 < p-value < 0.05@
qu.4.3.part.2.choice.2=0.05 < p-value < 0.10@
qu.4.3.part.2.choice.1=p-value > 0.10@
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qu.4.3.part.3.answer.2=No@
qu.4.3.part.3.answer.1=Yes@
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qu.4.3.question=<p>Consider the following summary statistics, taken on two independent&nbsp;random samples drawn from normally distributed populations:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>SAMPLE 1</strong></p>            </td>            <td>            <p align="center"><strong>SAMPLE 2</strong></p>            </td>        </tr>        <tr>            <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>$n1</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>$var1</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$var2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <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>$xbar1</mi></mrow></mstyle></math></td>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math></td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>Using the Welch approximate <em>t</em> procedure, test&nbsp;the null hypothesis&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>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><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>against the one-sided alternative <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msub><mi mathvariant='normal'>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Calculate&nbsp;the value for the <em>t</em> test statistic.</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp;&nbsp;Using&nbsp;a conservative estimate of the degrees of freedom (i.e. the lesser of <em>n<sub>1</sub> - 1</em>&nbsp;&nbsp;and <em>n<sub>2</sub> - 1</em>), the p-value is within which one of the following ranges?</span></p><p>&nbsp;</p><p><span>&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p>c)&nbsp; Is the null hypothesis rejected at the 10% level of significance?&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.4.4.mode=Inline@
qu.4.4.name=Calculate point estimate, margin of error for 99% confidence interval@
qu.4.4.comment=<p>a)&nbsp; A point estimate 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>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>&nbsp;is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$PE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The conservative estimate of the degrees of freedom is the smaller of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></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>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>, which in this case is $DFConservative.&nbsp; Therefore, for a 99% confidence interval for the difference in population means, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tAlpha2</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Using the Welch Approximate <em>t</em> procedure, 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>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n2</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SEWelch</mi></mrow></mstyle></math>.&nbsp; Therefore, the margin of error for a 99% confidence interval for the difference in population means is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SEWelch</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME</mi></mrow></mstyle></math>.</p>@
qu.4.4.editing=useHTML@
qu.4.4.solution=@
qu.4.4.algorithm=$n1=range(10,15);
$n2=range(10,15);
$s1=rand(4,6,2);
$var1=$s1^2;
$s2=rand(8,10,2);
$var2=$s2^2;
$xbar1=rand(70,80,4);
$xbar2=rand(75,85,4);
$PE=$xbar1-$xbar2;
$SEWelch=sqrt(($var1/$n1)+($var2/$n2));
$DFConservative=min($n1-1, $n2-1);
$tAlpha2=invstudentst($DFConservative, 0.995);
$ME=$tAlpha2*$SEWelch;@
qu.4.4.uid=f98a309a-7fd3-4418-90ea-8683207480d0@
qu.4.4.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.4.4.weighting=1,1,1@
qu.4.4.numbering=alpha@
qu.4.4.part.1.name=sro_id_1@
qu.4.4.part.1.answer.units=@
qu.4.4.part.1.numStyle=   @
qu.4.4.part.1.editing=useHTML@
qu.4.4.part.1.showUnits=false@
qu.4.4.part.1.err=0.0010@
qu.4.4.part.1.question=(Unset)@
qu.4.4.part.1.mode=Numeric@
qu.4.4.part.1.grading=toler_abs@
qu.4.4.part.1.negStyle=both@
qu.4.4.part.1.answer.num=$PE@
qu.4.4.part.2.name=sro_id_2@
qu.4.4.part.2.answer.units=@
qu.4.4.part.2.numStyle=   @
qu.4.4.part.2.editing=useHTML@
qu.4.4.part.2.showUnits=false@
qu.4.4.part.2.err=0.0010@
qu.4.4.part.2.question=(Unset)@
qu.4.4.part.2.mode=Numeric@
qu.4.4.part.2.grading=toler_abs@
qu.4.4.part.2.negStyle=both@
qu.4.4.part.2.answer.num=$tAlpha2@
qu.4.4.part.3.name=sro_id_3@
qu.4.4.part.3.answer.units=@
qu.4.4.part.3.numStyle=   @
qu.4.4.part.3.editing=useHTML@
qu.4.4.part.3.showUnits=false@
qu.4.4.part.3.err=0.0010@
qu.4.4.part.3.question=(Unset)@
qu.4.4.part.3.mode=Numeric@
qu.4.4.part.3.grading=toler_abs@
qu.4.4.part.3.negStyle=both@
qu.4.4.part.3.answer.num=$ME@
qu.4.4.question=<p>Consider two independent&nbsp;random samples of sizes <em>n<sub>1</sub> = $n1 </em>and <em>n<sub>2</sub> = $n2</em>, taken from two normally distributed populations.&nbsp; The sample standard deviations&nbsp;are calculated to be <em>s<sub>1</sub>= $s1</em>&nbsp; and <em>s<sub>2</sub>&nbsp; = $s2</em>, and the sample means&nbsp;are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi></mrow></mstyle></math>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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math>,&nbsp;respectively.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Calculate a point estimate for the difference in population 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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></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;&nbsp;Using the Welch approximate <em>t</em> procedure and&nbsp;a conservative estimate of the degrees of freedom (i.e. the lesser of <em>n<sub>1</sub> - 1</em>&nbsp;&nbsp;and <em>n<sub>2</sub> - 1</em>), what is the&nbsp;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>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>value for a 99% confidence interval?</span></p><p>&nbsp;</p><p>Round your response to at least&nbsp;3 decimal places.</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Calculate the margin of error for a 99% confidence interval.</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>@

qu.4.5.mode=Inline@
qu.4.5.name=Calculate test statistic, conclusion for one-sided hypothesis test (1)@
qu.4.5.comment=<p>a)&nbsp; Using the Welch Approximate <em>t </em>procedure, the <em>t</em> test statistic is determined through 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>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, and we can calculate the test statistic 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><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>$xbar2</mi></mrow></mfenced><mrow><msqrt><mrow><mfrac><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n2</mi></mrow></mfrac></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The conservative estimate of the degrees of freedom is the lesser of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></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>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>, which in this case is $DFConservative.&nbsp; Since the alternative hypothesis indicates that we are performing a one-sided, lower tailed test, the p-value is determined as the area under to the left of the test statistic, under a <em>t </em>distribution with $DFConservative degrees of freedom.&nbsp; Using computer software, we can find this area to be <em>p-value = $pvalue.</em></p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value is very small (less than 0.005), there is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis.&nbsp; Therefore, there is enough evidence to conclude that the mean of Population 1 is less than the mean of Population 2.</p>@
qu.4.5.editing=useHTML@
qu.4.5.solution=@
qu.4.5.algorithm=$n1=range(10,15);
$n2=range(10,15);
$s1=rand(1.1,3,3);
$var1=$s1^2;
$s2=rand(5,7,3);
$var2=$s2^2;
$xbar1=rand(-15,-10,3);
$xbar2=rand(-5,-0.01,3);
$PE=$xbar1-$xbar2;
$SEWelch=sqrt(($var1/$n1)+($var2/$n2));
$DFConservative=min($n1-1, $n2-1);
$tTest=$PE/$SEWelch;
$pvalue=studentst($DFConservative, $tTest);
condition:lt($pvalue,0.005);@
qu.4.5.uid=ced46d55-3706-4eee-8c10-78442d4972e6@
qu.4.5.info=  Course=Introductory Statistics;
  Topic=Inference for Two Sample Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.4.5.weighting=1,1,1@
qu.4.5.numbering=alpha@
qu.4.5.part.1.name=sro_id_1@
qu.4.5.part.1.answer.units=@
qu.4.5.part.1.numStyle=   @
qu.4.5.part.1.editing=useHTML@
qu.4.5.part.1.showUnits=false@
qu.4.5.part.1.err=0.01@
qu.4.5.part.1.question=(Unset)@
qu.4.5.part.1.mode=Numeric@
qu.4.5.part.1.grading=toler_abs@
qu.4.5.part.1.negStyle=both@
qu.4.5.part.1.answer.num=$tTest@
qu.4.5.part.2.name=sro_id_2@
qu.4.5.part.2.editing=useHTML@
qu.4.5.part.2.choice.5=p-value < 0.005@
qu.4.5.part.2.fixed=@
qu.4.5.part.2.choice.4=0.005 < p-value < 0.01@
qu.4.5.part.2.question=null@
qu.4.5.part.2.choice.3=0.01 < p-value < 0.05@
qu.4.5.part.2.choice.2=0.05 < p-value < 0.10@
qu.4.5.part.2.choice.1=p-value > 0.10@
qu.4.5.part.2.mode=Non Permuting Multiple Choice@
qu.4.5.part.2.display=vertical@
qu.4.5.part.2.answer=5@
qu.4.5.part.3.comment.3=@
qu.4.5.part.3.comment.2=@
qu.4.5.part.3.name=sro_id_3@
qu.4.5.part.3.comment.1=@
qu.4.5.part.3.editing=useHTML@
qu.4.5.part.3.choice.5=The results of the hypothesis test are invalid, since the assumptions of the Welch approximate t procedure were not met.@
qu.4.5.part.3.fixed=@
qu.4.5.part.3.choice.4=We can be completely certain the that means of the two populations are equal, as the p-value is very large.@
qu.4.5.part.3.question=null@
qu.4.5.part.3.choice.3=We can be completely certain that the mean of Population 1 is less than the mean of Population 2, as the p-value is very small.@
qu.4.5.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore no significant&nbsp;evidence that the two population means are different.@
qu.4.5.part.3.choice.1=There is sufficient evidence to reject the null hypothesis in favour of the alternative, that the mean of Population 1 is less than that of Population 2.@
qu.4.5.part.3.mode=Multiple Choice@
qu.4.5.part.3.display=vertical@
qu.4.5.part.3.comment.5=@
qu.4.5.part.3.comment.4=@
qu.4.5.part.3.answer=1@
qu.4.5.question=<p>Consider two independent&nbsp;random samples of sizes <em>n<sub>1</sub> = $n1 </em>and <em>n<sub>2</sub> = $n2</em>, taken from two normally distributed populations.&nbsp; The sample standard deviations&nbsp;are calculated to be <em>s<sub>1</sub>= $s1</em>&nbsp; and <em>s<sub>2</sub>&nbsp; = $s2</em>, and the sample means&nbsp;are <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar1</mi></mrow></mstyle></math>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>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar2</mi></mrow></mstyle></math>,&nbsp;respectively.&nbsp; Using this information, 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 mathvariant='normal'>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>against the one-sided alternative <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msub><mi>&mu;</mi><mrow><mn>1</mn></mrow></msub><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>&mu;</mi><mrow><mn>2</mn></mrow></msub></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>, using the Welch Approximate <em>t</em> Procedure.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Calculate&nbsp;the value for the <em>t</em> test statistic.</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp;&nbsp;Using&nbsp;a conservative estimate of the degrees of freedom (i.e. the lesser of <em>n<sub>1</sub> - 1</em>&nbsp;&nbsp;and <em>n<sub>2</sub> - 1</em>), the p-value is within which one of the following ranges?</span></p><p>&nbsp;</p><p><span>&nbsp;</span><2><span>&nbsp;</span>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>c)&nbsp; What is the most appropriate conclusion that can be made, at the 1% level of significance?</p><p>&nbsp;</p><p><span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.4.6.mode=Inline@
qu.4.6.name=Calculate Welch SE versus pooled standard deviation@
qu.4.6.comment=<p>a)&nbsp; Using the Welch Approximate <em>t</em> procedure, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>1</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>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mfrac><msubsup><mi>s</mi><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msubsup><mi>s</mi><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values leads to the calculation of the standard error of the difference in sample means 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><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup><mrow><mi>$n2</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SEWelch</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The estimate of the pooled variance is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><msub><mi>n</mi><mrow><mn>1</mn></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><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><msub><mi>n</mi><mrow><mn>2</mn></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><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values result in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><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><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><msup><mi>$s1</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$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><msup><mi>$s2</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$n2</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp2</mi></mrow></mstyle></math>, and therefore the estimate of the pooled standard deviation 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>s</mi><mrow><mi>p</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$sp2</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sp</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The more appropriate method to use in this situation is the Welch Approximate <em>t</em> procedure, since it is not known if the population variances are equal, and the sample sizes are very different from each other.</p>@
qu.4.6.editing=useHTML@
qu.4.6.solution=@
qu.4.6.algorithm=$n1=range(50,55);
$n2=range(8,13);
$s1=rand(4,5,3);
$s2=rand(6,7,3);
$sp2=(($n1-1)*$s1^2 + ($n2-1)*$s2^2)/($n1+$n2-2);
$sp=sqrt($sp2);
$SEWelch=sqrt(($s1^2/$n1)+($s2^2/$n2));@
qu.4.6.uid=3ae99876-dee0-45c4-a362-b530eb7d507b@
qu.4.6.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.4.6.weighting=1,1,1@
qu.4.6.numbering=alpha@
qu.4.6.part.1.name=sro_id_1@
qu.4.6.part.1.answer.units=@
qu.4.6.part.1.numStyle=   @
qu.4.6.part.1.editing=useHTML@
qu.4.6.part.1.showUnits=false@
qu.4.6.part.1.err=0.0010@
qu.4.6.part.1.question=(Unset)@
qu.4.6.part.1.mode=Numeric@
qu.4.6.part.1.grading=toler_abs@
qu.4.6.part.1.negStyle=both@
qu.4.6.part.1.answer.num=$SEWelch@
qu.4.6.part.2.name=sro_id_2@
qu.4.6.part.2.answer.units=@
qu.4.6.part.2.numStyle=   @
qu.4.6.part.2.editing=useHTML@
qu.4.6.part.2.showUnits=false@
qu.4.6.part.2.err=0.0010@
qu.4.6.part.2.question=(Unset)@
qu.4.6.part.2.mode=Numeric@
qu.4.6.part.2.grading=toler_abs@
qu.4.6.part.2.negStyle=both@
qu.4.6.part.2.answer.num=$sp@
qu.4.6.part.3.name=sro_id_3@
qu.4.6.part.3.editing=useHTML@
qu.4.6.part.3.fixed=@
qu.4.6.part.3.question=null@
qu.4.6.part.3.choice.2=The Pooled-Variance t Procedure@
qu.4.6.part.3.choice.1=The Welch Approximate t Procedure@
qu.4.6.part.3.mode=Multiple Choice@
qu.4.6.part.3.display=vertical@
qu.4.6.part.3.answer=1@
qu.4.6.question=<p>Consider the following summary statistics, obtained from independent&nbsp;samples drawn from two normally distributed populations:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>SAMPLE 1</strong></p>            </td>            <td>            <p align="center"><strong>SAMPLE 2</strong></p>            </td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n1</mi></mrow></mstyle></math></td>            <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>$n2</mi></mrow></mstyle></math></td>        </tr>        <tr>            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s1</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>s</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi></mrow></mstyle></math></td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Calculate&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>X</mi></mrow><mi>&macr;</mi></mover><mrow><mn>1</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>2</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>, using&nbsp;the&nbsp;Welch approximate <em>t</em>&nbsp;procedure.</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; Calculate the pooled standard deviation<em>,&nbsp;s<sub>p</sub>&nbsp;.</em></p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>c)&nbsp; Which method, the Welch approximate <em>t</em> procedure or the pooled-variance <em>t</em> procedure, is the most appropriate to use in this situation?</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><3><span>&nbsp;</span></span></p>@

qu.5.topic=Paired Difference Procedure@

qu.5.1.mode=Inline@
qu.5.1.name=Calculate point estimates, margin of error for 95% confidence interval@
qu.5.1.comment=<p>a)&nbsp; A point estimate for the mean difference is found&nbsp;by 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><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub><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>9</mn></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn>9</mn></mrow></mfrac></mrow></mrow></mstyle></math>, where <em>x<sub>i</sub> </em>is value for the difference.&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><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$meanDiff</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The standard error for the mean difference is found by dividing the standard deviation of the differences by the number of observations, such as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>.&nbsp; The standard deviation of the differences can be found by taking the square root of the variance, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>9</mn></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><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>8</mn></mrow></mfrac></mrow></msqrt></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Using the value found in part (a) 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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mstyle></math>, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sdDiff</mi></mrow></mstyle></math>.&nbsp; Therefore, the standard error is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>$sdDiff</mi></mrow><mrow><msqrt><mrow><mn>9</mn></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SEDiff</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The margin of error for a 95% confidence interval for the mean difference 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><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mfenced></mrow></mstyle></math>.&nbsp; The degrees of freedom for the <em>t</em> distribution is <em>n - 1 = 9 - 1 = 8, </em>and therefore for a 95% confidence interval, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>$tAlpha2</mi></mrow></mstyle></math>.&nbsp; Finally, the margin of error can be calculated as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SEDiff</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME</mi></mrow></mstyle></math>.</p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$Sample1=[rand(50,60,4),rand(50,60,4),rand(50,60,4),rand(50,60,4),rand(50,60,4),rand(50,60,4),rand(50,60,4),rand(50,60,4),rand(50,60,4)];
$Sample2=[rand(55,65,4),rand(55,65,4),rand(55,65,4),rand(55,65,4),rand(55,65,4),rand(55,65,4),rand(55,65,4),rand(55,65,4),rand(55,65,4)];
$Data=maple("
XList:=$Sample1-$Sample2:
X1:=Statistics[Mean](XList):
X2:=Statistics[StandardDeviation](XList):
XOut:=convert(XList, string):
XOut, X1, X2
");
$Differences=switch(0, $Data);
$meanDiff=switch(1, $Data);
$sdDiff=switch(2, $Data);
$SEDiff=$sdDiff/3;
$tAlpha2=invstudentst(8, 0.975);
$ME=$tAlpha2*$SEDiff;@
qu.5.1.uid=fd25bae3-4649-432b-adad-211d6aae6503@
qu.5.1.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.5.1.weighting=1,1,1@
qu.5.1.numbering=alpha@
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qu.5.1.part.1.numStyle=   @
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qu.5.1.part.1.err=0.0010@
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qu.5.1.part.2.err=0.0010@
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qu.5.1.part.2.grading=toler_abs@
qu.5.1.part.2.negStyle=both@
qu.5.1.part.2.answer.num=$SEDiff@
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qu.5.1.part.3.numStyle=   @
qu.5.1.part.3.editing=useHTML@
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qu.5.1.part.3.err=0.01@
qu.5.1.part.3.question=(Unset)@
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qu.5.1.part.3.negStyle=both@
qu.5.1.part.3.answer.num=$ME@
qu.5.1.question=<p>Consider the following set of random measurements, taken from a normally distributed population before and after a treatment was applied.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="500" align="center">    <tbody>        <tr>            <td><strong>Before Treatment</strong></td>            <td>$Sample1</td>        </tr>        <tr>            <td><strong>After Treatment</strong></td>            <td>$Sample2</td>        </tr>        <tr>            <td><strong>Difference</strong></td>            <td>$Differences</td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; Determine the point estimate for the mean difference.</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 standard error&nbsp;of the sample mean difference.</span></p><p>&nbsp;</p><p><span>Round your response to at least 3&nbsp;decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>c)&nbsp; What is the margin of error for a 95% confidence interval for the mean difference?</span></p><p>&nbsp;</p><p><span>Round&nbsp;your response to at least&nbsp;3 decimal places.</span></p><p><span>&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></span></p><p>&nbsp;</p>@

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qu.5.2.name=Calculate test statistic, p-value for one-sided hypothesis test (2)@
qu.5.2.comment=<p>a)&nbsp; The formula for the test statistic is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</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><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mfenced><mrow><mfrac><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></mrow></mfenced><mrow><mfrac><mi>$s</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are performing a one-sided, lower tailed test, which means that the p-value is the area to the left of the test statistic, under a <em>t </em>distribution with <em>$n - 1 = $DF</em> degrees of freedom.&nbsp; Using computer software, we can find this value to be exactly <em>p-value = $pvalue</em>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value found in part (b) is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis.</p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$n=range(12, 17);
$DF=$n-1;
$xbar=rand(-15, -10, 3);
$s=rand(40, 45, 3);
$SE=$s/sqrt($n);
$tTest=$xbar/$SE;
$pvalue=studentst($DF, $tTest);
condition:gt($pvalue,0.10);@
qu.5.2.uid=69382d0e-ad41-4cb1-b5cc-3ecaca5a15f4@
qu.5.2.info=  Course=Introductory Statistics;
  Topic=Inference for Two Sample Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.5.2.part.1.err=0.0010@
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qu.5.2.part.2.choice.5=p-value < 0.010@
qu.5.2.part.2.fixed=@
qu.5.2.part.2.choice.4=0.010 < p-value < 0.025@
qu.5.2.part.2.question=null@
qu.5.2.part.2.choice.3=0.025 < p-value < 0.05@
qu.5.2.part.2.choice.2=0.05 < p-value < 0.10@
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qu.5.2.part.3.credit.2=1.0@
qu.5.2.part.3.credit.1=0.0@
qu.5.2.question=<p>Consider the following summary statistics that were calculated on the <em><strong>difference</strong></em> between two dependent&nbsp;random samples, obtained from&nbsp;two normally distributed populations:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Summary Statistic</strong></p>            </td>        </tr>        <tr>            <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><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$n</mi></mrow></mstyle></math></p>            </td>        </tr>        <tr>            <td>            <p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><msub><mi>x</mi><mrow><mi></mi></mrow></msub></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math></p>            </td>        </tr>        <tr>            <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><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s</mi></mrow></mstyle></math></p>            </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><msub><mi>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></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><msub><mi>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; Calculate the value of the <em>t</em> test statistic.</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the range in which the p-value falls?</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Is there sufficient evidence, at the 10% level of significance, to reject the null hypothesis, in favour of the alternative hypothesis?</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.5.3.mode=Inline@
qu.5.3.name=Calculate 90% confidence interval@
qu.5.3.comment=<p>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><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>100</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>%</mi></mrow></mstyle></math>confidence interval for the mean difference is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub><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><mrow><mfrac><mrow><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow><mrow><msqrt><mrow><mi mathcolor='#800080'>n</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; The appropriate degrees of freedom for the <em>t</em> distribution is <em>$n - 1 = $DF</em>, and therefore for a 90% confidence interval <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>t</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tAlpha2</mi></mrow></mstyle></math>.&nbsp; Substituting in the remaing 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>$xbar</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&pm;</mo><mi>$tAlpha2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>$s</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>, which results in the interval <em>($LL, $UL).</em></p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$n=range(15, 20);
$DF=$n-1;
$xbar=rand(0.1, 0.6, 3);
$s=rand(0.01, 0.05, 3);
$SE=$s/sqrt($n);
$tAlpha2=invstudentst($DF, 0.95);
$ME=$tAlpha2*$SE;
$UL=$xbar + $ME;
$LL=$xbar - $ME;@
qu.5.3.uid=1a7df64b-b497-4c4c-9f89-5b59daa18a79@
qu.5.3.info=  Course=Introductory Statistics;
  Topic=Inference for Two Sample Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.5.3.weighting=1@
qu.5.3.numbering=alpha@
qu.5.3.part.1.editing=useHTML@
qu.5.3.part.1.question=(Unset)@
qu.5.3.part.1.name=sro_id_1@
qu.5.3.part.1.answer=($LL?0.001, $UL?0.001)@
qu.5.3.part.1.mode=Ntuple@
qu.5.3.question=<p>Consider the following summary statistics that were calculated on the <em><strong>difference</strong></em> between two dependent&nbsp;random samples, obtained from two&nbsp;normally distributed populations:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Summary Statistic</strong></p>            </td>        </tr>        <tr>            <td>            <p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$n</mi></mrow></mstyle></math></p>            </td>        </tr>        <tr>            <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><msub><mover><mrow><msub><mi>x</mi><mrow><mi></mi></mrow></msub></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math></p>            </td>        </tr>        <tr>            <td>            <p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s</mi></mrow></mstyle></math></p>            </td>        </tr>    </tbody></table></p><p>&nbsp;</p><p>Calculate a 90% confidence interval for the mean difference, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mstyle></math>.</p><p>&nbsp;</p><p>Enter your response in the format: <strong>(Lower limit, Upper limit)</strong> .</p><p>&nbsp;</p><p>Round each value to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.5.4.mode=Inline@
qu.5.4.name=Calculate test statistic, p-value for two-sided hypothesis test.@
qu.5.4.comment=<p>a)&nbsp; The formula for the <em>t</em> test statistic is given by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mfenced><mrow><mfrac><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></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><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mstyle></math>are the mean and standard deviation of the differences, respectively, 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><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mstyle></math>is the hypothesized difference.&nbsp; Using the values given, we can calculate the summary statistics 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><mi mathvariant='normal'>D</mi></mrow></msub><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>6</mn></mrow></munderover><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$meanDiff</mi></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>6</mn></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>5</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sdDiff</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>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>.&nbsp; Therefore, we get a value for the test statistic 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>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$meanDiff</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn></mrow></mfenced><mrow><mfrac><mrow><mi>$sdDiff</mi></mrow><mrow><msqrt><mrow><mn>6</mn></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Since the alternative hypothesis indicates that we are performing a two-sided test, the p-value is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$tTest</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math>, where <em>t</em> follows a <em>t</em> distribution with <em>6 - 1 = 5</em> degrees of freedom.&nbsp; Using computer software, we can find the area in the tail to be exactly $Tail, and therefore the p-value is <em>2 X $Tail = $pvalue</em>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; i)&nbsp; At the 5% level of significance, the p-value is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, and therefore there is insufficient evidence to reject the null hypothesis.</p>
<p>ii)&nbsp; Since the p-value is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis at the 10% level of significance.</p>@
qu.5.4.editing=useHTML@
qu.5.4.solution=@
qu.5.4.algorithm=$Sample1=[rand(6,8,3),rand(6,8,3),rand(6,8,3),rand(6,8,3),rand(6,8,3),rand(6,8,3)];
$Sample2=[rand(5,7,3),rand(5,7,3),rand(5,7,3),rand(5,7,3),rand(5.5,7.5,3),rand(5,7,3)];
$Data=maple("
XList:=$Sample1-$Sample2:
X1:=Statistics[Mean](XList):
X2:=Statistics[StandardDeviation](XList):
XOut:=convert(XList, string):
XOut, X1, X2
");
$Differences=switch(0, $Data);
$meanDiff=switch(1, $Data);
$sdDiff=switch(2, $Data);
$SEDiff=$sdDiff/sqrt(6);
$tTest=$meanDiff/$SEDiff;
$Tail=1-studentst(5, $tTest);
$pvalue=$Tail*2;
condition:gt($pvalue,0.05);
condition:lt($pvalue,0.10);@
qu.5.4.uid=487c43e5-f5b3-49af-8fee-de3fbc6cc3b3@
qu.5.4.info=  Course=Introductory Statistics;
  Topic=Inference for Two Sample Means, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
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qu.5.4.part.2.choice.5=p-value < 0.010@
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qu.5.4.part.2.choice.4=0.010 < p-value < 0.025@
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qu.5.4.question=<p>Consider the following set of random measurements, taken from a normally distributed population before and after a treatment was applied.</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="500" align="center">    <tbody>        <tr>            <td><strong>Before Treatment</strong></td>            <td>$Sample1</td>        </tr>        <tr>            <td><strong>After Treatment</strong></td>            <td>$Sample2</td>        </tr>        <tr>            <td><strong>Difference</strong></td>            <td>$Differences</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><msub><mi>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math>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><msub><mi>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0</mn></mrow></mstyle></math>.</p><p>&nbsp;</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the <em>t</em> test statistic?</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp;&nbsp;What is the range in which the p-value falls?</span></p><p>&nbsp;</p><p><span>&nbsp;<span>&nbsp;</span><2></span><span><span>&nbsp;</span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>c)&nbsp; Is the null hypothesis rejected at:</span></p><p><span>i)&nbsp; the 5% level of significance?&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></span></p><p>&nbsp;</p><p><span><span>ii)&nbsp; the 10% level of significance?&nbsp;&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span></span></span></p>@

qu.5.5.mode=Inline@
qu.5.5.name=Calculate test statistic, p-value for one-sided hypothesis test (1)@
qu.5.5.comment=<p>a)&nbsp; The formula for the <em>t</em> test statistic is given as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub></mrow></mfenced><mrow><mfrac><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we can calculate the test statistic 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><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$xbar</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn></mrow></mfenced><mrow><mfrac><mi>$s</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$tTest</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are performing a one-sided, upper tailed test.&nbsp; Therefore, the p-value is the area to the right of the test statistic, under a <em>t</em> distribution with <em>$n - 1 = $DF</em> degrees of freedom.&nbsp; Using computer software, we can find this area to be exactly <em>p-value = $pvalue</em>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since <em>p-value = $pvalue < <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.01</mn></mrow></mstyle></math>, </em>there is sufficient evidence to reject the null hypothesis in favour of the alternative hypothesis, at the 1% level of significance.</p>@
qu.5.5.editing=useHTML@
qu.5.5.solution=@
qu.5.5.algorithm=$n=range(4, 8);
$DF=$n-1;
$xbar=rand(0.1, 0.6, 3);
$s=rand(0.05, 0.09, 3);
$SE=$s/sqrt($n);
$tTest=$xbar/$SE;
$pvalue=1-studentst($DF, $tTest);
condition:lt($pvalue,0.01);@
qu.5.5.uid=bad797e8-e76d-45d3-9910-3359833afc68@
qu.5.5.info=  Course=Introductory Statistics;
  Topic=Inference for Two Sample Means, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
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qu.5.5.part.2.choice.5=p-value < 0.010@
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qu.5.5.part.2.choice.4=0.010 < p-value < 0.025@
qu.5.5.part.2.question=null@
qu.5.5.part.2.choice.3=0.025 < p-value < 0.05@
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qu.5.5.part.2.answer=5@
qu.5.5.part.3.grader=exact@
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qu.5.5.question=<p>Consider the following summary statistics that were calculated on the <em><strong>difference</strong></em> between two dependent&nbsp;random samples, obtained from&nbsp;two normally distributed populations:</p><p>&nbsp;</p><p><table border="1" cellspacing="1" cellpadding="1" width="200" align="center">    <tbody>        <tr>            <td>            <p align="center"><strong>Summary Statistic</strong></p>            </td>        </tr>        <tr>            <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><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$n</mi></mrow></mstyle></math></p>            </td>        </tr>        <tr>            <td>            <p align="center"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><msub><mi>x</mi><mrow><mi></mi></mrow></msub></mrow><mi>&macr;</mi></mover><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$xbar</mi></mrow></mstyle></math></p>            </td>        </tr>        <tr>            <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><msub><mi>s</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s</mi></mrow></mstyle></math></p>            </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><msub><mi>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></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><msub><mi>&mu;</mi><mrow><mi mathvariant='normal'>D</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; Calculate the value of the <em>t</em> test statistic.</p><p>&nbsp;</p><p>Round your response to at least 3 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the range in which the p-value falls?</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; Is there sufficient evidence, at the 1% level of significance, to reject the null hypothesis, in favour of the alternative hypothesis?</span></span></p><p>&nbsp;</p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

