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

qu.1.1.mode=Inline@
qu.1.1.name=Calculate point estimates, standard error and margin of error for 95% confidence interval@
qu.1.1.comment=<p>a)&nbsp; Point estimates of <em>p<sub>1</sub> </em>and <em>p<sub>2</sub></em>&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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x1</mi><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat1</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x2</mi><mrow><mi>$n2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat2</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To calculate the standard error for the difference in sample proportions, we can use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msqrt><mrow><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msqrt></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>q</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$qhat1</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$qhat2</mi></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>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><mrow><mi>$phat1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$qhat1</mi></mrow><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>$phat2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$qhat2</mi></mrow><mrow><mi>$n2</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; For a 95% confidence interval, the appropriate z-value is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1.96</mn></mrow></mstyle></math>.&nbsp; Therefore, the margin of error is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1.96</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME</mi></mrow></mstyle></math>.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$n1=range(100, 110);
$n2=range(105, 115);
$x1=range(30, 40);
$x2=range(40, 50);
$phat1=$x1/$n1;
$phat2=$x2/$n2;
$qhat1=1-$phat1;
$qhat2=1-$phat2;
$SE=sqrt(($phat1*$qhat1/$n1) + ($phat2*$qhat2/$n2));
$ME=1.96*$SE;@
qu.1.1.uid=fa73f92a-9e14-4d42-801c-069f4fe5e1aa@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Proportions, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.1.weighting=1,1,1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
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qu.1.1.part.1.numStyle=   @
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.err=0.0010@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=toler_abs@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$phat1@
qu.1.1.part.2.name=sro_id_2@
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qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.showUnits=false@
qu.1.1.part.2.err=0.0010@
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qu.1.1.part.2.grading=toler_abs@
qu.1.1.part.2.negStyle=both@
qu.1.1.part.2.answer.num=$phat2@
qu.1.1.part.3.name=sro_id_3@
qu.1.1.part.3.answer.units=@
qu.1.1.part.3.numStyle=   @
qu.1.1.part.3.editing=useHTML@
qu.1.1.part.3.showUnits=false@
qu.1.1.part.3.err=0.01@
qu.1.1.part.3.question=(Unset)@
qu.1.1.part.3.mode=Numeric@
qu.1.1.part.3.grading=toler_abs@
qu.1.1.part.3.negStyle=both@
qu.1.1.part.3.answer.num=$SE@
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qu.1.1.part.4.editing=useHTML@
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qu.1.1.part.4.grading=toler_abs@
qu.1.1.part.4.negStyle=both@
qu.1.1.part.4.answer.num=$ME@
qu.1.1.question=<p>Independent random samples of sizes <em>n<sub>1</sub> = $n1</em>&nbsp; and <em>n<sub>2</sub> = $n2</em>&nbsp;were taken from two populations.&nbsp; In the first sample, $x1 of the individuals met a certain criteria whereas in the second sample, $x2 of the individuals met the same criteria.</p><p>&nbsp;</p><p>a)&nbsp; What are the point estimates of <em>p<sub>1</sub> </em>and <em>p<sub>2</sub></em>, the true population proportion of individuals who meet this criteria in populations 1 and 2, respectivley?</p><p>&nbsp;</p><p>Round your responses to at least 3 decimal places.</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>b)&nbsp; Calculate the standard error of difference in sample proportions, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>, for a 95% confidence interval for the difference in population proportions.</span></span></p><p>&nbsp;</p><p><span><span>Round your response to at least 3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>c)&nbsp; What is the margin of error for a 95% confidence interval for the difference in population proportions?</span></span></span></p><p>&nbsp;</p><p><span><span><span>Round your response to at least&nbsp;3 decimal places.</span></span></span></p><p><span><span><span><span>&nbsp;</span><4><span>&nbsp;</span></span></span></span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Calculate standard error and margin of error for confidence interval@
qu.1.2.comment=<p>a)&nbsp; To calculate the standard error for the difference in sample proportions, we can use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msqrt><mrow><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub></mrow></mfenced><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></msqrt></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>q</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$qhat1</mi></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$qhat2</mi></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>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><mrow><mi>$phat1</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$qhat1</mi></mrow><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>$phat2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$qhat2</mi></mrow><mrow><mi>$n2</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; For a $ConfLevel % confidence interval, we need to determine the appropriate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>&nbsp;value.&nbsp; We need to find the value of <em>z</em> such that the area under the standard normal curve between +/- <em>z</em> is $ConfDecimal.&nbsp; Graphically, this becomes:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Using computer software, or approximating from a standard normal table, we can find this 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><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><mi>$zAlpha2</mi></mrow></mstyle></math>.&nbsp; Therefore, the margin of error is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$zAlpha2</mi><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.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$phat1=rand(0.4, 0.6, 2);
$percent1=$phat1*100;
$phat2=rand(0.3, 0.5, 2);
$percent2=$phat2*100;
$n1=range(200, 250);
$n2=range(200, 250);
$qhat1=1-$phat1;
$qhat2=1-$phat2;
$SE=sqrt(($phat1*$qhat1/$n1) + ($phat2*$qhat2/$n2));
$ConfLevel=range(70, 80);
$ConfDecimal=$ConfLevel/100;
$Quantile=($ConfDecimal/2)+0.5;
$zAlpha2=inverf($Quantile);
$zAlphaDisplay=decimal(3, $zAlpha2);
$ME=$zAlpha2*$SE;
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..-1*$zAlpha2, colour=blue): 
p2 := plot(f, x=-1*$zAlpha2..$zAlpha2, colour=blue, filled=true):
p3 := plot(f, x=$zAlpha2..3, colour=blue):
p4 := plots[textplot]([-1*$zAlphaDisplay, -0.05, `-$zAlphaDisplay`], color=blue):
p5 := plots[textplot]([$zAlphaDisplay, -0.05, `$zAlphaDisplay`], color=blue):
p6 := plots[textplot]([0, 0.15, `-$ConfDecimal-`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6}), plotoptions='width=350,height=350'
");@
qu.1.2.uid=931cb38c-cee6-4b01-9045-a3c5500f9198@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Proportions, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.2.weighting=1,1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.answer.units=@
qu.1.2.part.1.numStyle=   @
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.showUnits=false@
qu.1.2.part.1.err=0.01@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.mode=Numeric@
qu.1.2.part.1.grading=toler_abs@
qu.1.2.part.1.negStyle=both@
qu.1.2.part.1.answer.num=$SE@
qu.1.2.part.2.name=sro_id_2@
qu.1.2.part.2.answer.units=@
qu.1.2.part.2.numStyle=   @
qu.1.2.part.2.editing=useHTML@
qu.1.2.part.2.showUnits=false@
qu.1.2.part.2.err=0.01@
qu.1.2.part.2.question=(Unset)@
qu.1.2.part.2.mode=Numeric@
qu.1.2.part.2.grading=toler_abs@
qu.1.2.part.2.negStyle=both@
qu.1.2.part.2.answer.num=$ME@
qu.1.2.question=<p>Independent random samples of sizes <em>n<sub>1</sub> = $n1</em>&nbsp; and <em>n<sub>2</sub> = $n2</em>&nbsp;were taken from two populations.&nbsp; In the first sample,&nbsp;$percent1 % of the individuals met a certain criteria whereas in the second sample,&nbsp;$percent2 % of the individuals met the same criteria.</p><p>&nbsp;</p><p>&nbsp;<span><span>a)&nbsp; Calculate the standard error of difference in sample proportions, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>, for a $ConfLevel % confidence interval for the difference in population proportions.</span></span></p><p>&nbsp;</p><p><span><span>Round your response to at least 3 decimal places.</span></span></p><p><span><span><span>&nbsp;</span><1><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>b)&nbsp; What is the margin of error for a $ConfLevel % confidence interval for the difference in population proportions?</span></span></span></p><p>&nbsp;</p><p><span><span><span>Round your response to at least 3 decimal places.</span></span></span></p><p><span><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></span></p>@

qu.1.3.mode=Inline@
qu.1.3.name=Calculate test statistic, p-value for one-sided hypothesis test (1)@
qu.1.3.comment=<p>a)&nbsp; The formula for the <em>z</em> test statistic 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>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>p</mi></mrow><mi>&#x005e;</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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><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><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></msqrt></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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x1</mi><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat1</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x2</mi><mrow><mi>$n2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat2</mi></mrow></mstyle></math>.&nbsp; The pooled sample proportion 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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mi>X</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>X</mi><mrow><mn>2</mn></mrow></msub></mrow></mfenced><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></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$x2</mi></mrow></mfenced><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></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Pooledphat</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>p</mi><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Pooledqhat</mi></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>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>$phat1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$phat2</mi></mrow></mfenced><mrow><msqrt><mrow><mi>$Pooledphat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Pooledqhat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi>$n2</mi></mrow></mfrac></mrow></mfenced></mrow></msqrt></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>= $zTest.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are performing a one-sided, lower-tailed&nbsp;test.&nbsp; Therefore, to find the p-value we must determine the area under the standard normal curve&nbsp;to the left&nbsp;of the test statistic.</p>
<p>Using computer software, or approximating with a standard&nbsp;normal table,&nbsp;we can find this area to be $pvalue.</p>
<p>&nbsp;</p>
<p>c)&nbsp;&nbsp;When <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, the p-value is less than the level of significance.&nbsp; Therefore, there is sufficient evidence to reject the null hypothesis at the 10% level of significance.&nbsp; At the 5% level of significance, the p-value is greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>.&nbsp; Therefore, there is insufficient evidence to reject the null hypothesis at the 5% level of significance.</p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$n1=range(300, 310);
$n2=range(300, 310);
$x1=range(90, 100);
$x2=range(105, 115);
$phat1=$x1/$n1;
$phat2=$x2/$n2;
$Pooledphat=($x1 + $x2)/($n1 + $n2);
$Pooledqhat=1-$Pooledphat;
$SE=sqrt($Pooledphat*$Pooledqhat*(1/$n1 + 1/$n2));
$zTest=($phat1-$phat2)/$SE;
$pvalue=erf($zTest);
condition:not(eq($phat1,$phat2));
condition:lt($pvalue,0.10);
condition:gt($pvalue,0.05);@
qu.1.3.uid=1bc24292-8dff-4bd1-8ec7-7612de483fc6@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Proportions, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.3.part.1.answer.num=$zTest@
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qu.1.3.part.2.answer.num=$pvalue@
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qu.1.3.part.3.fixed=@
qu.1.3.part.3.choice.4=There is insufficient evidence to reject the null hypothesis at both the 5% and 10% levels of significance.@
qu.1.3.part.3.question=null@
qu.1.3.part.3.choice.3=There is sufficient evidence to reject the null hypothesis at the 5% level of significance, but insufficienct evidence to reject the null hypothesis&nbsp;at the 10% level of significance.@
qu.1.3.part.3.choice.2=There is sufficient evidence to reject the null hypothesis at both the 10% and 5% levels of significance.@
qu.1.3.part.3.choice.1=There is sufficient evidence to reject the null hypothesis at the 10% level of significance, but insufficient evidence to reject the null hypothesis at the 5% level of significance.@
qu.1.3.part.3.mode=Multiple Choice@
qu.1.3.part.3.display=vertical@
qu.1.3.part.3.answer=1@
qu.1.3.question=<p>Independent random samples of sizes <em>n<sub>1</sub> = $n1</em>&nbsp; and <em>n<sub>2</sub> = $n2</em>&nbsp;were taken from two populations.&nbsp; In the first sample, $x1 of the individuals met a certain criteria whereas in the second sample, $x2 of the individuals met the same criteria.</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>p</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>p</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>versus the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>p</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>p</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp;&nbsp;Calculate&nbsp;the <em>z</em>&nbsp; test statistic, testing the null hypothesis that the population proportions are equal.</p><p>&nbsp;</p><p>Round your response 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; What is the approximate value of the p-value?</span></span></p><p>&nbsp;</p><p><span>&nbsp;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><span>c)&nbsp;&nbsp;What is the appropriate conclusion that can be made?</span></span></span>&nbsp;</p><p><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p><p>&nbsp;</p>@

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qu.1.4.name=Definitions 1: Inference for Two Population Proportions@
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qu.1.4.solution=@
qu.1.4.algorithm=@
qu.1.4.uid=64cca917-560d-4678-9932-5e9025b4a12d@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Proportions;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.4.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>Note that there may be more than one correct answer; select all that are true.</p>@
qu.1.4.answer=1, 2@
qu.1.4.choice.1=The 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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math> is approximately normal for large sample sizes.@
qu.1.4.choice.2=In hypothesis testing for the difference between two population proportions, a pooled estimate of the sample proportion is used when calculating the test statistic.@
qu.1.4.choice.3=When estimating a confidence interval for the difference in population proportions, the standard error of the difference in sample proportions is dependent upon <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>p</mi><mrow><msub><mn>0</mn><mrow><mn>1</mn></mrow></msub></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><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>p</mi><mrow><msub><mn>0</mn><mrow><mn>2</mn></mrow></msub></mrow></msub></mrow></mstyle></math> (i.e. the hypothesized population proportions).@
qu.1.4.choice.4=An estimate for the pooled sample proportion 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><mfrac><msub><mi>X</mi><mrow><mn>1</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><msub><mi>X</mi><mrow><mn>2</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mstyle></math>, or in other words, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math>.@
qu.1.4.choice.5=One of the required assumptions for the inference procedures on two proportions to be valid is that the populations from which the two samples are drawn are dependent on each other.@
qu.1.4.fixed=@

qu.1.5.mode=Inline@
qu.1.5.name=Calculate pooled proportion, standard error, test statistic@
qu.1.5.comment=<p>a)&nbsp; The pooled sample proportion 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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mi>X</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>X</mi><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mrow><msub><mi>n</mi><mrow><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$x2</mi></mrow></mfenced><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></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Pooledphat</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To calculate the standard error for the difference in sample proportions, we can use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><msqrt><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><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><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></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><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$Pooledphat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Pooledqhat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi>$n2</mi></mrow></mfrac></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SE</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; The formula for the <em>z</em> test statistic 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>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>p</mi></mrow><mi>&#x005e;</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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><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><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></msqrt></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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x1</mi><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat1</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x2</mi><mrow><mi>$n2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat2</mi></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>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>$phat1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$phat2</mi></mrow></mfenced><mrow><msqrt><mrow><mi>$Pooledphat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Pooledqhat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><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></mfenced></mrow></msqrt></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>= $zTest.</p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$n1=range(400, 410);
$n2=range(305, 315);
$x1=range(100, 120);
$x2=range(120, 140);
$phat1=$x1/$n1;
$phat2=$x2/$n2;
$Pooledphat=($x1 + $x2)/($n1 + $n2);
$Pooledqhat=1-$Pooledphat;
$SE=sqrt($Pooledphat*$Pooledqhat*(1/$n1 + 1/$n2));
$zTest=($phat1-$phat2)/$SE;@
qu.1.5.uid=5ef021b3-1148-4bc0-870b-2c74623c3c68@
qu.1.5.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Proportions, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.5.weighting=1,1,1@
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qu.1.5.part.1.numStyle=   @
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qu.1.5.part.1.showUnits=false@
qu.1.5.part.1.err=0.0010@
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qu.1.5.part.1.negStyle=both@
qu.1.5.part.1.answer.num=$Pooledphat@
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qu.1.5.part.2.err=0.01@
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qu.1.5.part.2.grading=toler_abs@
qu.1.5.part.2.negStyle=both@
qu.1.5.part.2.answer.num=$SE@
qu.1.5.part.3.name=sro_id_3@
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qu.1.5.part.3.numStyle=   @
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qu.1.5.part.3.grading=toler_abs@
qu.1.5.part.3.negStyle=both@
qu.1.5.part.3.answer.num=$zTest@
qu.1.5.question=<p>Random samples of sizes <em>n<sub>1</sub> = $n1</em>&nbsp; and <em>n<sub>2</sub> = $n2</em>&nbsp;were taken from two independent populations.&nbsp; In the first sample, $x1 of the individuals met a certain criteria whereas in the second sample, $x2 of the individuals met the same criteria.</p><p>&nbsp;</p><p>a)&nbsp; What is the&nbsp;value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>, the pooled sample proportion?</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><span>b)&nbsp; Calculate the standard error of the&nbsp;difference in sample proportions, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced></mrow></mstyle></math>, using the pooled estimate of the sample proportion.</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><2><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>c)&nbsp; Calculate the <em>z</em> test statistic, testing the null hypothesis that the population proportions are equal.</span></span></span></p><p>&nbsp;</p><p><span><span><span>Round your response to at least&nbsp;3 decimal places.</span></span></span></p><p><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p>@

qu.1.6.mode=Multiple Selection@
qu.1.6.name=Definitions 2: Inference for Two Population Proportions@
qu.1.6.comment=@
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qu.1.6.solution=@
qu.1.6.algorithm=@
qu.1.6.uid=ab62e8a8-9b2f-49ff-92e8-9ac883b27131@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Proportions;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.6.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>Note that there may be more than one correct answer; select all that are true.</p>@
qu.1.6.answer=1, 2, 3@
qu.1.6.choice.1=Assuming the two populations are independent, then the variance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mrow></mstyle></math> is equal to the variance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub></mrow></mstyle></math> plus the variance of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.@
qu.1.6.choice.2=The mean of 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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>is</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>p</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msub><mi>p</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>@
qu.1.6.choice.3=The standard error of the difference in sample proportions for confidence intervals is different from that for hypothesis testing.@
qu.1.6.choice.4=If one of the sample sizes is small, a two-sample t procedure can be used for inference on two population proportions.@
qu.1.6.choice.5=In hypothesis testing on two proportions, we are often interested in testing whether or not the two sample proportions are equal to each other.@
qu.1.6.fixed=@

qu.1.7.mode=Inline@
qu.1.7.name=Calculate test statistic, p-value for one-sided hypothesis test (2)@
qu.1.7.comment=<p>a)&nbsp; The formula for the <em>z</em> test statistic 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>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>p</mi></mrow><mi>&#x005e;</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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><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><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></msqrt></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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x1</mi><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat1</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x2</mi><mrow><mi>$n2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat2</mi></mrow></mstyle></math>.&nbsp; The pooled sample proportion 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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mi>X</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>X</mi><mrow><mn>2</mn></mrow></msub></mrow></mfenced><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></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$x2</mi></mrow></mfenced><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></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Pooledphat</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>p</mi><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Pooledqhat</mi></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>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>$phat1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$phat2</mi></mrow></mfenced><mrow><msqrt><mrow><mi>$Pooledphat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Pooledqhat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><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></mfenced></mrow></msqrt></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>= $zTest.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are performing a one-sided, upper-tailed&nbsp;test.&nbsp; Therefore, to find the p-value we must determine the area under the standard normal curve&nbsp;to the&nbsp;right&nbsp;of the test statistic.</p>
<p>Using computer software, or approximating with a standard&nbsp;normal table,&nbsp;we can find this area to be $pvalue.</p>
<p>&nbsp;</p>
<p>c)&nbsp;&nbsp;When <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, the p-value is&nbsp;greater than the level of significance.&nbsp;&nbsp;Therefore, there is insufficient evidence to reject the null hypothesis at the 10% level of significance, and therefore no significant evidence that the population proportions are not equal to each other.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$n1=range(200, 210);
$n2=range(200, 210);
$x1=range(170, 180);
$x2=range(175, 185);
$phat1=$x1/$n1;
$phat2=$x2/$n2;
$Pooledphat=($x1 + $x2)/($n1 + $n2);
$Pooledqhat=1-$Pooledphat;
$SE=sqrt($Pooledphat*$Pooledqhat*(1/$n1 + 1/$n2));
$zTest=($phat1-$phat2)/$SE;
$pvalue=1-erf($zTest);
condition:not(eq($phat1,$phat2));@
qu.1.7.uid=60bab869-72a3-4105-824c-6a2d390c4831@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Proportions, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.7.part.1.answer.num=$zTest@
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qu.1.7.part.2.answer.num=$pvalue@
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qu.1.7.question=<p>Independent random samples of sizes <em>n<sub>1</sub> = $n1</em>&nbsp; and <em>n<sub>2</sub> = $n2</em>&nbsp;were taken from two populations.&nbsp; In the first sample, $x1 of the individuals met a certain criteria whereas in the second sample, $x2 of the individuals met the same criteria.</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>p</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>p</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>versus the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>p</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>p</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp;&nbsp;Calculate the <em>z</em>&nbsp; test statistic, testing the null hypothesis that the population proportions are equal.</p><p>&nbsp;</p><p>Round your response 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; What is the approximate value of the p-value?</span></span></p><p>&nbsp;</p><p><span>&nbsp;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><span>c)&nbsp;&nbsp;Is there any evidence, at the 10% level of significance,&nbsp;to indicate that <em>p<sub>1</sub> </em>and <em>p<sub>2</sub> </em>are not equal to each other?&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p>@

qu.1.8.mode=Inline@
qu.1.8.name=Calculate test statistic, p-value for two-sided hypothesis test@
qu.1.8.comment=<p>a)&nbsp; The formula for the <em>z</em> test statistic 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>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>p</mi></mrow><mi>&#x005e;</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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><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><mfrac><mn>1</mn><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></msqrt></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>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x1</mi><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat1</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><mrow><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x2</mi><mrow><mi>$n2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat2</mi></mrow></mstyle></math>.&nbsp; The pooled sample proportion 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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><msub><mi>X</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>X</mi><mrow><mn>2</mn></mrow></msub></mrow></mfenced><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></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mfenced open='(' close=')' separators=','><mrow><mi>$x1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$x2</mi></mrow></mfenced><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></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Pooledphat</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mi>p</mi><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Pooledqhat</mi></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>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>$phat1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$phat2</mi></mrow></mfenced><mrow><msqrt><mrow><mi>$Pooledphat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$Pooledqhat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>$n1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi>$n2</mi></mrow></mfrac></mrow></mfenced></mrow></msqrt></mrow></mfrac></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>= $zTest.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are performing a two-sided test.&nbsp; Therefore, the p-value is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$zTest</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; We must determine the area in the tail,&nbsp;under the standard normal curve, and then&nbsp;multiply this area by 2.&nbsp; Because the test statistic is a large negative number, the area&nbsp;in the tail of the&nbsp;standard normal distribution&nbsp;is very small.&nbsp; Using computer software, we can find this area to be $Tail, and therefore <em>p-value&nbsp;= 2 x $Tail = $pvalue</em>.&nbsp; If you were approximating the p-value from a standard normal table, we would say the p-value is approximately 0.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value is very very small, we can conclude that at the 5% level of significance, there is strong evidence to reject the null hypothesis that the population proportions are the same, in favour of the alternative hypothesis that they are not the same.</p>@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$n1=range(400, 410);
$n2=range(300, 310);
$x1=range(100, 120);
$x2=range(150, 170);
$phat1=$x1/$n1;
$phat2=$x2/$n2;
$Pooledphat=($x1 + $x2)/($n1 + $n2);
$Pooledqhat=1-$Pooledphat;
$SE=sqrt($Pooledphat*$Pooledqhat*(1/$n1 + 1/$n2));
$zTest=($phat1-$phat2)/$SE;
$Tail=erf($zTest);
$pvalue=2*$Tail;@
qu.1.8.uid=0b78d00c-6ad7-4b98-8fcb-1d62a76a83c9@
qu.1.8.info=  Course=Introductory Statistics;
  Topic=Inference for Two Population Proportions, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.8.part.3.choice.2=There is no evidence to reject the null hypothesis, and therefore no significant&nbsp;evidence that the population proportions are not equal to each other.@
qu.1.8.part.3.choice.1=There is strong evidence&nbsp;to reject&nbsp;the null hypothesis, in favour of the alternative that the population proportions are not equal to each other.@
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qu.1.8.question=<p>Independent random samples of sizes <em>n<sub>1</sub> = $n1</em>&nbsp; and <em>n<sub>2</sub> = $n2</em>&nbsp;were taken from two populations.&nbsp; In the first sample, $x1 of the individuals met a certain criteria whereas in the second sample, $x2 of the individuals met the same criteria.</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>p</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>p</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>versus the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>p</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><msub><mi>p</mi><mrow><mn>2</mn></mrow></msub></mrow></mstyle></math>.</p><p>&nbsp;</p><p>a)&nbsp; What is the value of the <em>z</em>&nbsp; test statistic, testing the null hypothesis that the population proportions are equal?</p><p>&nbsp;</p><p>Round your response 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; What is the approximate value of the p-value?</span></span></p><p>&nbsp;</p><p><span>&nbsp;</span><2><span>&nbsp;</span>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>c)&nbsp; What conclusion can be made, at the 5% level of significance?</span></span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p><p>&nbsp;</p>@

