qu.1.topic=Inference for Single Population Proportion@

qu.1.1.mode=Inline@
qu.1.1.name=Determine minium sample size for 95%, 99% confidence, conservative@
qu.1.1.comment=<p>a)&nbsp; In order to deterimine the minimum sample size required, 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>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>q</mi></mrow></mstyle></math>.&nbsp; To estimate <em>p</em> within <em>m = $m</em>, with 95% confidence indicates that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1.96</mn></mrow></mstyle></math>.&nbsp; Because there is no information on the value of <em>p</em>, we can use the conservative value of <em>p = 0.5, q = 0.5</em>.&nbsp; Substituting in the values, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1.96</mn><mrow><mi>$m</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.5</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n95Decimal</mi></mrow></mstyle></math>.&nbsp; Since the sample size must by a whole number, we round this value up to <em>$n95Round.</em></p>
<p>&nbsp;</p>
<p>b)&nbsp; To estimate the minimum sample size required to be within the same margin of error as part (a), but with 99% confidence, we can use the same formula as above but replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>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>2.575</mn></mrow></mstyle></math>.&nbsp; This results in <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>2.575</mn><mrow><mi>$m</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.5</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n99Decimal</mi></mrow></mstyle></math>.&nbsp; Again, this value must be rounded to <em>n = $n99Round.</em></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$m=rand(0.01, 0.05, 1);
$n95Decimal=(1.96/$m)^2*0.5*0.5;
$n99Decimal=(2.575/$m)^2*0.5*0.5;
$Values=maple("
X1:=ceil($n95Decimal):
X2:=ceil($n99Decimal):
X1, X2
");
$n95Round=switch(0, $Values);
$n99Round=switch(1, $Values);@
qu.1.1.uid=df7c6d4d-bf2f-4049-a38b-02306a1c6002@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion, Determining Sample Size;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.1.weighting=1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.answer.units=@
qu.1.1.part.1.numStyle=thousands   @
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=exact_value@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$n95Round@
qu.1.1.part.2.name=sro_id_2@
qu.1.1.part.2.answer.units=@
qu.1.1.part.2.numStyle=thousands   @
qu.1.1.part.2.editing=useHTML@
qu.1.1.part.2.showUnits=false@
qu.1.1.part.2.question=(Unset)@
qu.1.1.part.2.mode=Numeric@
qu.1.1.part.2.grading=exact_value@
qu.1.1.part.2.negStyle=both@
qu.1.1.part.2.answer.num=$n99Round@
qu.1.1.question=<p>Determine the minimum sample size required in order to estimate <em>p</em>, the population proportion, to within $m, with:</p><p>&nbsp;</p><p>a)&nbsp;95% confidence.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; 99% confidence.</span></p><p>&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span></p>@

qu.1.2.mode=Inline@
qu.1.2.name=Determine minium sample size for 90% confidence@
qu.1.2.comment=<p>a)&nbsp; In order to deterimine the minimum sample size required, 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>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow><mrow><mi>m</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>q</mi></mrow></mstyle></math>.&nbsp; To estimate <em>p</em> within <em>m = $m</em>, with 90% confidence indicates that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1.645</mn></mrow></mstyle></math>.&nbsp;&nbsp;Using the information on the value of <em>p</em>, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mn>1.645</mn></mrow><mrow><mi>$m</mi></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$q</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$nDecimal</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.&nbsp; Since the sample size must by a whole number, we round this value up to <em>$nRound.</em></p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$m=rand(0.01, 0.05, 1);
$p=rand(0.5, 0.8, 2);
$q=1-$p;
$nDecimal=(1.645/$m)^2*$p*$q;
$nRound=maple("
X:=ceil($nDecimal):
X
");@
qu.1.2.uid=458af892-c8c1-4141-b86b-f844be09ccd4@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion, Determining Sample Size;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.2.weighting=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=thousands   @
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.showUnits=false@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.mode=Numeric@
qu.1.2.part.1.grading=exact_value@
qu.1.2.part.1.negStyle=both@
qu.1.2.part.1.answer.num=$nRound@
qu.1.2.question=<p>Determine the minimum sample size required in order to estimate <em>p</em>, the population proportion, to within $m with 90% confidence, when a&nbsp;previous study has shown that&nbsp;&nbsp;<em>p</em>&nbsp;is approximately&nbsp;$p.&nbsp; Use this value in your formula for determining sample size.</p><p>&nbsp;</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.3.mode=Multiple Selection@
qu.1.3.name=Definition 1: Inference for Single Population Proportion@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=@
qu.1.3.uid=c4620475-0b72-4b9c-be56-1d694fc08979@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.3.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>Note that there may be more than one correct answer; select all that are true.</p>@
qu.1.3.answer=1, 2, 3@
qu.1.3.choice.1=Generally speaking, the confidence interval procedures are valid if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><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><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math> are both greater than or equal to 15.@
qu.1.3.choice.2=As the desired margin of error decreases, then the required minimum sample size will increase.@
qu.1.3.choice.3=The formula for standard error of the sample proportion differs between confidence intervals and hypothesis testing, in that for confidence intervals we use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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='0.0em' 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><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow></mstyle></math>, and in hypothesis testing we use <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mn>0</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='bold' fontweight='bold' lspace='0.0em' rspace='0.0em'>and</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>q</mi><mrow><mn>0</mn></mrow></msub></mrow></mstyle></math>.@
qu.1.3.choice.4=In order for the inference procedures for proportions to be valid, the population from which the random sample is drawn must be normally distributed.@
qu.1.3.choice.5=When determining the minimum sample size, if the values of p and q are not given, we should use the "worst-case scenario" values of p = 0.1 and q = 0.9 @
qu.1.3.fixed=@

qu.1.4.mode=Inline@
qu.1.4.name=Calculate point estimate, standard error@
qu.1.4.comment=<p>a)&nbsp; The point estimate of <em>p</em>, the population 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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>, the sample proportion, 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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>x</mi><mrow><mi>n</mi></mrow></mfrac></mrow></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><mi>$x</mi><mrow><mi>$n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The standard error of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>&nbsp;is given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><mfrac><mrow><mi>$phat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$qhat</mi></mrow><mrow><mi>$n</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SE</mi></mrow></mstyle></math></p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$n=range(80, 100);
$x=range(40, 50);
$phat=$x/$n;
$qhat=1-$phat;
$SE=sqrt(($phat*$qhat)/$n);@
qu.1.4.uid=1441fef9-bef7-447b-8174-4a626578244c@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.4.weighting=1,1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
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qu.1.4.part.1.numStyle=   @
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.showUnits=false@
qu.1.4.part.1.err=0.0010@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.mode=Numeric@
qu.1.4.part.1.grading=toler_abs@
qu.1.4.part.1.negStyle=both@
qu.1.4.part.1.answer.num=$phat@
qu.1.4.part.2.name=sro_id_2@
qu.1.4.part.2.answer.units=@
qu.1.4.part.2.numStyle=   @
qu.1.4.part.2.editing=useHTML@
qu.1.4.part.2.showUnits=false@
qu.1.4.part.2.err=0.01@
qu.1.4.part.2.question=(Unset)@
qu.1.4.part.2.mode=Numeric@
qu.1.4.part.2.grading=toler_abs@
qu.1.4.part.2.negStyle=both@
qu.1.4.part.2.answer.num=$SE@
qu.1.4.question=<p>A&nbsp;simple random sample of size $n is&nbsp;taken from a population,&nbsp;and $x of the individuals meet a specified criteria.</p><p>&nbsp;</p><p>a)&nbsp; What is a point estimate for <em>p</em>, the proportion of individuals in the population that meet the criteria?</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&nbsp;the value&nbsp;of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced></mrow></mstyle></math>, the standard error of the sample proportion.</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>@

qu.1.5.mode=Multiple Selection@
qu.1.5.name=Definition 2: Inference for Single Population Proportion@
qu.1.5.comment=@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=@
qu.1.5.uid=e77be405-629f-46f5-9828-fd1881eed931@
qu.1.5.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.5.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.1.5.answer=1, 2@
qu.1.5.choice.1=For hypothesis testing on the population proportion, the test statistic that is calculated approximately follows a standard normal distribution, assuming the null hypothesis is true and the sample size is large.@
qu.1.5.choice.2=The true standard error of the sample proportion requires that we know p, the population proportion.  However, since this value is often unknown, we replace with the sample proportion, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>, or the hypothesized proportion, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mn>0</mn></mrow></msub></mrow></mstyle></math>.@
qu.1.5.choice.3=As a rough guideline, the hypothesis testing procedures are valid if <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>p</mi><mrow><mn>0</mn></mrow></msub><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>15</mn><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><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>q</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>15</mn></mrow></mstyle></math>.@
qu.1.5.choice.4=Inference procedures for proportions will never work if the proportion is near 0 or 1.@
qu.1.5.choice.5=Assuming all else is held constant, and 90% confidence interval for p will be wider than a 95% confidence interval.@
qu.1.5.fixed=@

qu.1.6.mode=Inline@
qu.1.6.name=Calculate hypotheses, test statistic, p-value for one-sided hypothesis test (2)@
qu.1.6.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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><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><msub><mi>p</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mfrac><mrow><msub><mi>p</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>q</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>, where <em>p<sub>0</sub> </em>is the hypothesized proportion, and <em>q<sub>0 </sub>= 1 - p<sub>0</sub>.&nbsp; </em>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>$phat</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0.8</mn></mrow></mfenced><mrow><msqrt><mrow><mfrac><mrow><mn>0.8</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.2</mn></mrow><mrow><mi>$n</mi></mrow></mfrac></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><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; Because the alternate hypothesis is one-sided and upper-tailed, the p-value is the area&nbsp;to the&nbsp;right of the test statistic.&nbsp; Using computer software, or approximating from a standard normal table, we can find this area to be $pvalue.&nbsp; Graphically, the p-value is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p align="center">&nbsp;</p>
<p align="left">c)&nbsp; Because the p-value = $pvalue is&nbsp;greater than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is insufficient evidence to reject the null hypothesis at the 10% level of significance.&nbsp; Therefore, there is no&nbsp;evidence that the population proportion is&nbsp;not 0.8.</p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$n=range(1000, 1020);
$x=range(800, 820);
$phat=$x/$n;
$qhat=1-$phat;
$SE=sqrt((0.8*0.2)/$n);
$zTest=($phat-0.8)/$SE;
$zDisplay=decimal(2, $zTest);
$pvalue=1-erf($zTest);
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..$zTest, colour=blue): 
p2 := plot(f, x=$zTest..3, colour=blue, filled=true): 
p3 := plots[textplot]([$zTest, -0.05, `$zDisplay`], color=blue):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");
condition:gt($pvalue,0.10);@
qu.1.6.uid=df2c2c19-1f36-4e12-960f-82e0ede37201@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.6.weighting=1,1,1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.answer.units=@
qu.1.6.part.1.numStyle=   @
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.showUnits=false@
qu.1.6.part.1.err=0.0010@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.mode=Numeric@
qu.1.6.part.1.grading=toler_abs@
qu.1.6.part.1.negStyle=both@
qu.1.6.part.1.answer.num=$zTest@
qu.1.6.part.2.name=sro_id_2@
qu.1.6.part.2.answer.units=@
qu.1.6.part.2.numStyle=   @
qu.1.6.part.2.editing=useHTML@
qu.1.6.part.2.showUnits=false@
qu.1.6.part.2.err=0.01@
qu.1.6.part.2.question=(Unset)@
qu.1.6.part.2.mode=Numeric@
qu.1.6.part.2.grading=toler_abs@
qu.1.6.part.2.negStyle=both@
qu.1.6.part.2.answer.num=$pvalue@
qu.1.6.part.3.name=sro_id_3@
qu.1.6.part.3.editing=useHTML@
qu.1.6.part.3.fixed=@
qu.1.6.part.3.question=null@
qu.1.6.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population proportion is not 0.8.@
qu.1.6.part.3.choice.1=There is sufficient evidence to reject the null hypothesis, in favour of the alternative that the population proportion is&nbsp;greater than 0.8.@
qu.1.6.part.3.mode=Multiple Choice@
qu.1.6.part.3.display=vertical@
qu.1.6.part.3.answer=2@
qu.1.6.question=<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><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.8</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><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0.8</mn></mrow></mstyle></math>, when $x individuals in a random sample of $n have a characteristic of interest.&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><span>a)&nbsp; Calculate the value of the <em>z</em> test statistic, for testing the null hypothesis that the population proportion is 0.8.</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; Estimate the corresponding p-value for the above test statistic.</span></span></p><p>&nbsp;</p><p><span><span>Round your&nbsp;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>c)&nbsp; What conclusion can be made, at the 10% level of significance?</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><3><span>&nbsp;</span></span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Calculate z value, margin of error for confidence interval@
qu.1.7.comment=<p>a)&nbsp; To find the&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></mrow></mstyle></math>value for a $ConfLevel % confidence interval, we need to deterimine the value of <em>z</em> such that under a standard normal curve, the area between +/- <em>z</em> is $ConfDec.&nbsp; Graphically, this becomes:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p>&nbsp;</p>
<p>Using computer software (or approximating with a standard normal table), we can find that&nbsp;this value&nbsp;is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>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>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To find the margin of error, we need to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>and&nbsp;the standard error of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>, 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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><msqrt><mrow><mfrac><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover></mrow></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msqrt></mrow></mrow></mstyle></math>.&nbsp; Substituting in 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><mi>x</mi><mrow><mi>n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x</mi><mrow><mi>$n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat</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><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><mrow><mi>p</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><mi>$phat</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$qhat</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><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mfrac><mrow><mi>$phat</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$qhat</mi></mrow><mrow><mi>$n</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SE</mi></mrow></mstyle></math>.&nbsp; Finally, we can find 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><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><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></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>
<p>&nbsp;</p>
<p>c)&nbsp; In order for the confidence interval procedures to be valid, we need to confirm if the guidelines 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>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>15</mn></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><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>15</mn></mrow></mstyle></math>are met.&nbsp; In this case, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&sdot;</mo><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$phat</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$nphat</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><mi>n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$qhat</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$nqhat</mi></mrow></mstyle></math>, which does not meet the guidelines.&nbsp; Therefore,&nbsp;it was <strong>not</strong> reasonable to use&nbsp;confidence interval procedures.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$n=range(10, 15);
$x=range(4,9);
$phat=$x/$n;
$qhat=1-$phat;
$nphat=$n*$phat;
$nqhat=$n*$qhat;
$ConfLevel=range(50,60);
$ConfDec=$ConfLevel/100;
$Quantile=($ConfDec/2)+0.5;
$zAlpha2=inverf($Quantile);
$zAlpha2T=decimal(2, $zAlpha2);
$SE=sqrt(($phat*$qhat)/$n);
$ME=$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=$zAlpha2..3, colour=blue): 
p4 := plots[textplot]([-1*$zAlpha2, -0.05, `-$zAlpha2T`], color=blue):
p5 := plots[textplot]([$zAlpha2, -0.05, `$zAlpha2T`], color=blue):
p6 := plots[textplot]([0, 0.15, `-$ConfDec-`], color=black):
plots[display]({p1,p2,p3,p4,p5,p6}), plotoptions='width=350,height=350'
");@
qu.1.7.uid=84fca838-c36f-4037-9701-2b9a54febfb8@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.7.part.1.answer.num=$zAlpha2@
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qu.1.7.question=<p>Suppose a random sample of size $n, of which $x have a certain criteria, is drawn from a population.</p><p>&nbsp;</p><p>a)&nbsp; If we were to use confidence interval procedures based on the normal distribution, what is the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub></mrow></mstyle></math>&nbsp;value for a $ConfLevel % confidence interval for <em>p</em>, the population proportion?</p><p>&nbsp;</p><p>Round your response to at least&nbsp;2 decimal places.</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What is the margin of error for a $ConfLevel % confidence interval for <em>p</em>?<br /></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>c)&nbsp; Was it reasonable to use the confidence interval procedures in this&nbsp;situation?&nbsp;&nbsp;<span>&nbsp;</span><3></span><span><span>&nbsp;</span>&nbsp;</span></p>@

qu.1.8.mode=Inline@
qu.1.8.name=Calculate hypotheses, test statistic, p-value for two-sided hypothesis test@
qu.1.8.comment=<p>a)&nbsp; Hypothesis testing is always conducted on the population parameter, which in this case is the population proportion, <em>p</em>.&nbsp; There is also the indication for a two-sided alternative hypothesis, 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><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>p</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0.3</mn></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><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><msub><mi>p</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mfrac><mrow><msub><mi>p</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>q</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>, where <em>p<sub>0</sub> </em>is the hypothesized proportion, and <em>q<sub>0 </sub>= 1 - p<sub>0</sub>.&nbsp; </em>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>$phat</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0.3</mn></mrow></mfenced><mrow><msqrt><mrow><mfrac><mrow><mn>0.3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.7</mn></mrow><mrow><mi>$n</mi></mrow></mfrac></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$zTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; Because the alternative hypothesis is two-sided, the p-value is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi>Z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$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>, which we can calculate as twice the area in the upper tail.&nbsp; Using computer software, or approximating from a standard normal table, we can find the area in the tail of the distribution&nbsp;to be $Tail.&nbsp; Therefore, the p-value is 2&nbsp;x&nbsp;$Tail&nbsp;= $pvalue.&nbsp; Graphically, the p-value is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>@
qu.1.8.editing=useHTML@
qu.1.8.solution=@
qu.1.8.algorithm=$n=range(400, 420);
$x=range(130, 140);
$phat=$x/$n;
$qhat=1-$phat;
$SE=sqrt((0.3*0.7)/$n);
$zTest=($phat-0.3)/$SE;
$zDisplay=decimal(2, $zTest);
$Tail=1-erf($zTest);
$pvalue=2*$Tail;
condition:gt($pvalue,0.10);
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-3..-1*$zTest, colour=blue, filled=true): 
p2 := plot(f, x=-1*$zTest..$zTest, colour=blue): 
p3 := plot(f, x=$zTest..3, colour=blue, filled=true):
p4 := plots[textplot]([-1*$zTest, -0.05, `-$zDisplay`], color=blue):
p5 := plots[textplot]([$zTest, -0.05, `$zDisplay`], color=blue):
plots[display]({p1,p2,p3,p4,p5}), plotoptions='width=350,height=350'
");@
qu.1.8.uid=891c1c9a-1d79-48af-9a35-53678e09f58b@
qu.1.8.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.8.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>p</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>0.3</mn><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>@
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qu.1.8.part.2.answer.num=$zTest@
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qu.1.8.part.3.grading=toler_abs@
qu.1.8.part.3.negStyle=both@
qu.1.8.part.3.answer.num=$pvalue@
qu.1.8.question=<p>A random sample of size $n is taken from a population.&nbsp; Within this sample, $x individuals carry a certain characteristic.&nbsp; Test the null hypothesis that the true proportion of individuals in the population with this characteristic is 0.3, against the two-sided alternative hypothesis.</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 value of the <em>z</em> test statistic.</span></p><p>&nbsp;</p><p><span>Round your response to at least 3 decimal places.</span></p><p><span><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span>c)&nbsp; What is the corresponding p-value for the test statistic found in part (b)?</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.1.9.mode=Inline@
qu.1.9.name=Calculate test statistic, p-value, conclusion for one-sided hypothesis test (1)@
qu.1.9.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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><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><msub><mi>p</mi><mrow><mn>0</mn></mrow></msub></mrow></mfenced><mrow><msqrt><mrow><mfrac><mrow><msub><mi>p</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>q</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>, where <em>p<sub>0</sub> </em>is the hypothesized proportion, and <em>q<sub>0 </sub>= 1 - p<sub>0</sub>.&nbsp; </em>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>$phat</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0.5</mn></mrow></mfenced><mrow><msqrt><mrow><mfrac><mrow><mn>0.5</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mn>0.5</mn></mrow><mrow><mi>$n</mi></mrow></mfrac></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; Because the alternative hypothesis is one-sided and lower-tailed, the p-value is the area&nbsp;to the&nbsp;left of the test statistic.&nbsp; Using computer software, or approximating from a standard normal table, we can find this area to be $pvalue.&nbsp; Graphically, the p-value is represented as:</p>
<p>&nbsp;</p>
<p align="center">$p</p>
<p align="center">&nbsp;</p>
<p align="left">c)&nbsp; Because the p-value = $pvalue is less than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is sufficient evidence to reject the null hypothesis at the 5% level of significance.&nbsp; Therefore, there is evidence that the population proportion is less than 0.5.</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$n=range(210, 220);
$x=range(85, 95);
$phat=$x/$n;
$qhat=1-$phat;
$SE=sqrt((0.5*0.5)/$n);
$zTest=($phat-0.5)/$SE;
$zDisplay=decimal(2, $zTest);
$pvalue=erf($zTest);
$p=plotmaple("
f := Statistics[PDF](Normal(0, 1),x): 
p1 := plot(f, x=-4..$zTest, colour=blue, filled=true): 
p2 := plot(f, x=$zTest..4, colour=blue): 
p3 := plots[textplot]([$zTest, -0.05, `$zDisplay`], color=blue):
plots[display]({p1,p2,p3}), plotoptions='width=350,height=350'
");
condition:lt($pvalue,0.05);@
qu.1.9.uid=6061cbb2-dcb3-4d4f-82c1-7d7eb4bc7c96@
qu.1.9.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.9.part.1.err=0.0010@
qu.1.9.part.1.question=(Unset)@
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qu.1.9.part.1.negStyle=both@
qu.1.9.part.1.answer.num=$zTest@
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qu.1.9.part.2.fixed=@
qu.1.9.part.2.choice.4=p-value < 0.05@
qu.1.9.part.2.question=null@
qu.1.9.part.2.choice.3=0.05 < p-value < 0.10@
qu.1.9.part.2.choice.2=0.10 < p-value < 0.50@
qu.1.9.part.2.choice.1=p-value > 0.50@
qu.1.9.part.2.mode=Non Permuting Multiple Choice@
qu.1.9.part.2.display=vertical@
qu.1.9.part.2.answer=4@
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qu.1.9.part.3.editing=useHTML@
qu.1.9.part.3.fixed=@
qu.1.9.part.3.question=null@
qu.1.9.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population proportion is not 0.5.@
qu.1.9.part.3.choice.1=There is sufficient evidence to reject the null hypothesis, in favour of the alternative that the population proportion is less than 0.5.@
qu.1.9.part.3.mode=Multiple Choice@
qu.1.9.part.3.display=vertical@
qu.1.9.part.3.answer=1@
qu.1.9.question=<p>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><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0.5</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><mi>p</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0.5</mn></mrow></mstyle></math>, when $x individuals in a random sample of $n have a characteristic of interest.&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p><span>a)&nbsp; Calculate the value of the <em>z</em> test statistic, for testing the null hypothesis that the population proportion is 0.5.</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; The p-value falls within which one of the following ranges:</span></span></p><p>&nbsp;</p><p><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>c)&nbsp; What conclusion can be made, at the 5% level of significance?</span></p><p>&nbsp;</p><p><span><span>&nbsp;</span><3><span>&nbsp;</span></span></p>@

qu.1.10.mode=Inline@
qu.1.10.name=Calculate margin of error for 90%, 95% confidence intervals@
qu.1.10.comment=<p>a)&nbsp; In order to calculate the margin of error for a 90% confidence interval for <em>p</em>, we first need to calculate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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 separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover></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><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced></mrow></mstyle></math>.&nbsp; Using the values of <em>x</em> = $x and <em>n</em> = $n, we get <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>$x</mi><mrow><mi>$n</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$phat</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><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><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$qhat</mi></mrow></mstyle></math>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><mfrac><mrow><mi>$phat</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$qhat</mi></mrow><mrow><mi>$n</mi></mrow></mfrac></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$SE</mi></mrow></mstyle></math>.&nbsp; For a 90% confidence interval, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1.645</mn></mrow></mstyle></math>, therefore the margin of error is calculated as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>ME</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>z</mi><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1.645</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$SE</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ME90</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; For a 95% confidence interval, we can use the values from 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><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mover><mrow><mi>q</mi></mrow><mi>&#x005e;</mi></mover></mrow></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><mi>SE</mi><mfenced open='(' close=')' separators=','><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mfenced></mrow></mstyle></math>, however in this case <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><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>$ME95</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$n=range(50, 70);
$x=range(20, 30);
$phat=$x/$n;
$qhat=1-$phat;
$SE=sqrt(($phat*$qhat)/$n);
$ME90=1.645*$SE;
$ME95=1.96*$SE;@
qu.1.10.uid=5707ad64-4bf1-489f-8d6e-35d9a3c3da07@
qu.1.10.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Proportion, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.10.weighting=1,1@
qu.1.10.numbering=alpha@
qu.1.10.part.1.name=sro_id_1@
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qu.1.10.part.1.numStyle=   @
qu.1.10.part.1.editing=useHTML@
qu.1.10.part.1.showUnits=false@
qu.1.10.part.1.err=0.01@
qu.1.10.part.1.question=(Unset)@
qu.1.10.part.1.mode=Numeric@
qu.1.10.part.1.grading=toler_abs@
qu.1.10.part.1.negStyle=both@
qu.1.10.part.1.answer.num=$ME90@
qu.1.10.part.2.name=sro_id_2@
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qu.1.10.part.2.numStyle=   @
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qu.1.10.part.2.grading=toler_abs@
qu.1.10.part.2.negStyle=both@
qu.1.10.part.2.answer.num=$ME95@
qu.1.10.question=<p>In a simple random sample of size $n, taken from a population, $x of the individuals met a specified criteria.</p><p>&nbsp;</p><p>a)&nbsp; What is the margin of error for a 90% confidence interval for <em>p</em>, the population 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>b)&nbsp; What is the margin of error for a 95% confidence interval for <em>p?</em></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>@

