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

qu.1.1.mode=Inline@
qu.1.1.name=Calculate degrees of freedom, Chi-square values for 95% confidence interval@
qu.1.1.comment=<p>a)&nbsp; The degrees of freedom for a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution are <em>n - 1</em>.&nbsp; Therefore, the appropriate degrees of freedom here are <em>$n - 1 = $df</em>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; For a 95% confidence interval, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>.&nbsp; Therefore, 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><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>0.025</mn></mrow><mrow><mn>2</mn></mrow></msubsup></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><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>0.975</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>.&nbsp; For the first value, we need the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;value, at $df degrees of freedom, that has an area above it of 0.025.&nbsp; Using computer software, or approximating with a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution table, we can find this value to be $ChiAlpha2.&nbsp; For the second value, we need the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>value, at $df degrees of freedom, with an area above it of 0.975.&nbsp; Again, using computer software (or approximating from a table), we can find this value to be $Chi1minusAlpha2.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$s2=range(20, 25, 3);
$n=range(15, 20);
$df=$n-1;
$ChiValues=maple("
UL:=Statistics[Quantile](ChiSquare($df), 0.025):
LL:=Statistics[Quantile](ChiSquare($df), 0.975):
UL, LL
");
$Chi1minusAlpha2=switch(0, $ChiValues);
$ChiAlpha2=switch(1, $ChiValues);@
qu.1.1.uid=5f7cd495-720c-4ae0-86a8-9dd78ae676f9@
qu.1.1.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Variance, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
qu.1.1.weighting=1,1,1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
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qu.1.1.part.1.numStyle=   @
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qu.1.1.part.1.showUnits=false@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.mode=Numeric@
qu.1.1.part.1.grading=exact_value@
qu.1.1.part.1.negStyle=both@
qu.1.1.part.1.answer.num=$df@
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qu.1.1.part.2.question=(Unset)@
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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=$ChiAlpha2@
qu.1.1.part.3.name=sro_id_3@
qu.1.1.part.3.answer.units=@
qu.1.1.part.3.numStyle=   @
qu.1.1.part.3.editing=useHTML@
qu.1.1.part.3.showUnits=false@
qu.1.1.part.3.err=0.0010@
qu.1.1.part.3.question=(Unset)@
qu.1.1.part.3.mode=Numeric@
qu.1.1.part.3.grading=toler_abs@
qu.1.1.part.3.negStyle=both@
qu.1.1.part.3.answer.num=$Chi1minusAlpha2@
qu.1.1.question=<p>A random sample of size <em>$n</em> is taken from a normally distributed population, and a sample variance of <em>$s2</em> is calculated.</p><p>If we are interested in creating a 95% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>, the population variance, then:</p><p>&nbsp;</p><p>a)&nbsp; What is the appropriate degrees of freedom for the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution?</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span>b)&nbsp; What are 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><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>&nbsp;values?</span></p><p>&nbsp;</p><p><span>Round your responses to at least 3 decimal places.</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p>&nbsp;</p><p><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span>&nbsp;</span></span></span></p>@

qu.1.2.mode=Multiple Selection@
qu.1.2.name=Definitions 1: Inference for a Single Population Variance@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=@
qu.1.2.uid=5a64c711-9f72-4729-b443-335682f8148e@
qu.1.2.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Variance;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.2.question=<p>Which of the following statements are true?</p>
<p>&nbsp;</p>
<p>Note that there may be more than one correct answer; select all that are true.</p>@
qu.1.2.answer=1, 2@
qu.1.2.choice.1=In order for the hypothesis testing procedures for population variance to be valid, the population from which the sample is drawn must be normally distributed.@
qu.1.2.choice.2=If the null hypothesis is true, then the statistic <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math> follows a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> distribution with n - 1 degrees of freedom.@
qu.1.2.choice.3=If a 95% confidence interval for the population variance contains 0, there is sufficient evidence to reject the null hypothesis at the 5% level of significance, in favour of the two-sided alternative.@
qu.1.2.choice.4=To find 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> values for a confidence interval for the population variance, the appropriate degrees of freedom for the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> distribution are n.@
qu.1.2.fixed=@

qu.1.3.mode=Inline@
qu.1.3.name=Calculate Chi-square values, upper and lower limits for 95% confidence interval@
qu.1.3.comment=<p>a)&nbsp; For a 95% confidence interval, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>.&nbsp; Therefore, 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><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>0.025</mn></mrow><mrow><mn>2</mn></mrow></msubsup></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><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>0.975</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>.&nbsp; For the first value, we need the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;value, at $df degrees of freedom, that has an area above it of 0.025.&nbsp; Using computer software, or approximating with a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution table, we can find this value to be $ChiAlpha2.&nbsp; For the second value, we need the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>value, at $df degrees of freedom, with an area above it of 0.975.&nbsp; Again, using computer software (or approximating from a table), we can find this value to be $Chi1minusAlpha2.</p>
<p>&nbsp;</p>
<p>b)&nbsp; A <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&alpha;</mi></mrow></mfenced><mn>100</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>%</mi></mrow></mstyle></math>&nbsp;confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>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><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>.&nbsp; Therefore, the <em>lower limit</em> of the 95% confidence interval 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><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$s2</mi></mrow><mrow><mi>$ChiAlpha2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Lower</mi></mrow></mstyle></math>, and the <em>upper limit</em> of the 95% confidence interval is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$s2</mi></mrow><mrow><mi>$Chi1minusAlpha2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Upper</mi></mrow></mstyle></math>.&nbsp; Therefore, the 95% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;is <em>($Lower, $Upper)</em>.</p>@
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qu.1.3.solution=@
qu.1.3.algorithm=$s2=range(15, 20, 3);
$n=range(10, 15);
$df=$n-1;
$ChiValues=maple("
UL:=Statistics[Quantile](ChiSquare($df), 0.025):
LL:=Statistics[Quantile](ChiSquare($df), 0.975):
UL, LL
");
$Chi1minusAlpha2=switch(0, $ChiValues);
$ChiAlpha2=switch(1, $ChiValues);
$Upper=($n-1)*$s2/$Chi1minusAlpha2;
$Lower=($n-1)*$s2/$ChiAlpha2;@
qu.1.3.uid=ad8f17b2-9cd2-4e75-bb62-0e785ba670d0@
qu.1.3.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Variance, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
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qu.1.3.question=<p>A random sample of size <em>$n</em> is taken from a normally distributed population, and a sample variance of <em>$s2</em> is calculated.</p><p>If we are interested in creating a 95% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>, the population variance, then:</p><p>&nbsp;</p><p><span>a)&nbsp; What are 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><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>&nbsp;values?</span></p><p>&nbsp;</p><p><span>Round your responses to at&nbsp;least&nbsp;3 decimal places.</span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>b)&nbsp; What are the values for the lower and upper limits of the confidence intervals?</span></span></span></p><p>&nbsp;</p><p><span><span><span>Enter your values separately, in the spaces provided.</span></span></span></p><p>&nbsp;</p><p><span><span><span>Round each of your&nbsp;responses to at least 3 decimal places.</span></span></span></p><p><span><span><span>Lower limit =&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></p><p>&nbsp;</p><p><span><span><span>Upper limit =&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span>&nbsp;</span></span></span></p>@

qu.1.4.mode=Inline@
qu.1.4.name=Calculate sample variance, upper and lower limits for 90% confidence interval@
qu.1.4.comment=<p>a)&nbsp; To&nbsp;calculate the sample variance, we can use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, and with a little algebra, 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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&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><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>.&nbsp; Therefore, we need to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>0.05</mn></mrow><mrow><mn>2</mn></mrow></msubsup></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><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>0.95</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>.&nbsp; For the first value, we need the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;value, at&nbsp;5 degrees of freedom, that has an area above it of 0.05.&nbsp; Using computer software, or approximating with a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution table, we can find this value to be $ChiAlpha2.&nbsp; For the second value, we need the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>value, at&nbsp;5 degrees of freedom, with an area above it of 0.95.&nbsp; Again, using computer software (or approximating from a table), we can find this value to be $Chi1minusAlpha2.</p>
<p>A <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&alpha;</mi></mrow></mfenced><mn>100</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>%</mi></mrow></mstyle></math>&nbsp;confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>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><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>.&nbsp; Therefore, the <em>lower limit</em> of the 90% confidence interval 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><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$s2</mi></mrow><mrow><mi>$ChiAlpha2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Lower</mi></mrow></mstyle></math>, and the <em>upper limit</em> of the 90% confidence interval is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$s2</mi></mrow><mrow><mi>$Chi1minusAlpha2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Upper</mi></mrow></mstyle></math>.&nbsp; Therefore, the 90% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;is <em>($Lower, $Upper)</em>.</p>@
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$ChiValues=maple("
S1:=Statistics[Variance]($Data):
UL:=Statistics[Quantile](ChiSquare(5), 0.05):
LL:=Statistics[Quantile](ChiSquare(5), 0.95):
UL, LL, S1
");
$Chi1minusAlpha2=switch(0, $ChiValues);
$ChiAlpha2=switch(1, $ChiValues);
$s2=switch(2, $ChiValues);
$Upper=(6-1)*$s2/$Chi1minusAlpha2;
$Lower=(6-1)*$s2/$ChiAlpha2;@
qu.1.4.uid=1bff61df-7939-4fa5-bac6-197e588a9916@
qu.1.4.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Variance, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
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qu.1.4.part.2.answer=($Lower?0.01, $Upper?0.01)@
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qu.1.4.question=<p>The following values are obtained from a random sample that was drawn from a normally distributed population.</p><p>&nbsp;</p><p align="center">$Data</p><p>If we are interested in creating a 90% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>, the population variance, then:</p><p>&nbsp;</p><p><span>a)&nbsp;&nbsp;Calculate <em>s<sup>2</sup></em>, the sample variance.</span></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><span>b)&nbsp; What are the values for the lower and upper limits of the confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>?</span></span></span></p><p>&nbsp;</p><p><span><span><span>Enter your response in the format: <strong>(Lower limit, Upper limit) </strong>.</span></span></span></p><p>&nbsp;</p><p><span><span><span>Round each of your&nbsp;values&nbsp;to at least 3 decimal places before entering them.</span></span></span></p><p><span><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></span></p>@

qu.1.5.mode=Inline@
qu.1.5.name=Calculate test statistic, range of p-value for two-sided hypothesis test@
qu.1.5.comment=<p>a)&nbsp; The formula for the test statistic is given as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>&sigma;</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></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><msubsup><mrow><mi>&sigma;</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>&nbsp;is the hypothesized population variance.&nbsp; Substituting in the appropriate values, the test statistic calculates 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$s2</mi></mrow><mrow><mn>15</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ChiTest</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are conducting a two-sided test.&nbsp; Therefore, to find the p-value, we need to calculate&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>P</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mi>$ChiTest</mi></mrow></mfenced></mrow></mstyle></math>,&nbsp;&nbsp;which is twice&nbsp;the area to the right of the test statistic, under the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution with $df degrees of freedom.&nbsp; Using computer software, we can find the exact area to the right of the test statistic to be <em>$UpperTail</em>,&nbsp;which means the p-value is<em> 2 X $UpperTail = $pvalue</em>.&nbsp;</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value is larger than <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.05</mn></mrow></mstyle></math>, there is&nbsp;no significant evidence against the null hypothesis, and therefore the&nbsp;null hypothesis is not rejected, at the 5% level of significance.&nbsp;&nbsp;</p>@
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qu.1.5.algorithm=$s2=rand(15.1, 18.0, 3);
$n=range(10, 15);
$df=$n-1;
$ChiTestRaw=($n-1)*$s2/15.0;
$ChiTest=numfmt("#.00000", $ChiTestRaw);
$LowerTail=maple("
X:=Statistics[CDF](ChiSquare($df), $ChiTest):
X
");
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$pvalue=$UpperTail*2;
condition:gt($pvalue,0.10);@
qu.1.5.uid=208b2ce0-7bfd-49c3-86d7-5ba42df59a39@
qu.1.5.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Variance, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
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qu.1.5.part.2.choice.5=p-value < 0.01@
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qu.1.5.part.2.choice.4=0.01 < p-value < 0.025@
qu.1.5.part.2.question=null@
qu.1.5.part.2.choice.3=0.025 < p-value < 0.05@
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qu.1.5.part.2.choice.1=p-value > 0.10@
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qu.1.5.part.3.fixed=@
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qu.1.5.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore no significant&nbsp;evidence that the population variance is not 15.@
qu.1.5.part.3.choice.1=There is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis that the population variance is not 15.@
qu.1.5.part.3.mode=Multiple Choice@
qu.1.5.part.3.display=vertical@
qu.1.5.part.3.answer=2@
qu.1.5.question=<p>A random sample of size <em>$n</em> is taken from a normally distributed population, and a sample variance of <em>$s2</em> is calculated.</p><p>Use this information to test the null hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>15</mn></mrow></mstyle></math>&nbsp;against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&ne;</mo><mn>15</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p><p>&nbsp;</p><p><span>a)&nbsp; What is the value of the test statistic?</span></p><p>&nbsp;</p><p><span>Round your response to at least&nbsp;3 decimal places.</span></p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>b)&nbsp; The p-value falls within which one of the following ranges:</span></span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span><span>c)&nbsp; What conclusion can be made, at the 5% level of significance?</span></span></span></span></p><p>&nbsp;</p><p><span><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></span></p>@

qu.1.6.mode=Inline@
qu.1.6.name=Calculate sample variance, test statistic, conclusion for one-sided hypothesis test (2)@
qu.1.6.comment=<p>a)&nbsp; In order to calculate the test statistic, we first must calculate the sample variance.&nbsp; This can be done using the formula:&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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we 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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi></mrow></mstyle></math>.&nbsp; The formula for the test statistic is given as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>&sigma;</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></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><msubsup><mrow><mi>&sigma;</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>&nbsp;is the hypothesized population variance.&nbsp; Substituting in the appropriate values, the test statistic calculates 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mn>5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$s2</mi></mrow><mrow><mn>120</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ChiTest</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are conducting a one-sided, lower-tailed&nbsp;test.&nbsp; Therefore, to find the p-value, we need to calculate the area to the left of the test statistic, under the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution&nbsp;with 4 degrees of freedom.&nbsp; Using computer software, we can find the exact&nbsp;p-value to be <em>$pvalue</em>.&nbsp;</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value 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='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is no&nbsp;significant evidence against the null hypothesis.&nbsp; Therefore the&nbsp;null hypothesis is not rejected at the 10% level of significance, and there is&nbsp;no significant evidence that the population variance is not 120.</p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$Data=[rand(5,10,4),rand(35,40,4),rand(20,25,4),rand(25,30,4),rand(15,20,4)];
$Values=maple("
S1:=Statistics[Variance]($Data):
CHISQ:=(5-1)*(S1)/120;
X:=Statistics[CDF](ChiSquare(4), CHISQ):
S1, CHISQ, X
");
$s2=switch(0, $Values);
$ChiTest=switch(1, $Values);
$pvalue=switch(2, $Values);
condition:gt($pvalue,0.50);@
qu.1.6.uid=99963f8e-a36c-4f5e-8ba4-36ad0aa4ce40@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Variance, 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=0.01@
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=$ChiTest@
qu.1.6.part.2.name=sro_id_2@
qu.1.6.part.2.editing=useHTML@
qu.1.6.part.2.choice.5=p-value < 0.01@
qu.1.6.part.2.fixed=@
qu.1.6.part.2.choice.4=0.01 < p-value < 0.025@
qu.1.6.part.2.question=null@
qu.1.6.part.2.choice.3=0.025 < p-value < 0.05@
qu.1.6.part.2.choice.2=0.05 < p-value < 0.10@
qu.1.6.part.2.choice.1=p-value > 0.10@
qu.1.6.part.2.mode=Non Permuting Multiple Choice@
qu.1.6.part.2.display=vertical@
qu.1.6.part.2.answer=1@
qu.1.6.part.3.name=sro_id_3@
qu.1.6.part.3.editing=useHTML@
qu.1.6.part.3.fixed=@
qu.1.6.part.3.question=null@
qu.1.6.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population variance is not 120.@
qu.1.6.part.3.choice.1=There is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis that the population variance is&nbsp;less than&nbsp;120.@
qu.1.6.part.3.mode=Multiple Choice@
qu.1.6.part.3.display=vertical@
qu.1.6.part.3.answer=2@
qu.1.6.question=<p>The following observations are obtained when a random sample is drawn from a normally distributed population:</p><p>&nbsp;</p><p align="center">$Data</p><p>&nbsp;</p><p>Use this information to test the null hypothesis&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>120</mn></mrow></mstyle></math>&nbsp;against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>120</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p><p>&nbsp;</p><p><span>a)&nbsp; What is the value of the test statistic?</span></p><p>&nbsp;</p><p><span>Round your response to at least&nbsp;3 decimal places.</span></p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>b)&nbsp; The p-value falls within which one of the following ranges:</span></span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span><span>c)&nbsp; What conclusion can be made, at the 10% level of significance?</span></span></span></span></p><p>&nbsp;</p><p><span><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Calculate sample standard deviation, 90% confidence interval for standard deviation@
qu.1.7.comment=<p>a)&nbsp; To estimate the sample standard deviation, we&nbsp;first use the formula to calculate sample variance:&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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, and with a little algebra, 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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi></mrow></mstyle></math>, and therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>s</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>$s2</mi></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; To find a confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi></mrow></mstyle></math>, we first need to find the confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>, and then take the square-root of the upper and lower confidence bounds.</p>
<p>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><mi>&alpha;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>.&nbsp; Therefore, we need to find&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>0.05</mn></mrow><mrow><mn>2</mn></mrow></msubsup></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><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>0.95</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>.&nbsp; For the first value, we need the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;value, at&nbsp;5 degrees of freedom, that has an area above it of 0.05.&nbsp; Using computer software, or approximating with a <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution table, we can find this value to be $ChiAlpha2.&nbsp; For the second value, we need the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>value, at&nbsp;5 degrees of freedom, with an area above it of 0.95.&nbsp; Again, using computer software (or approximating from a table), we can find this value to be $Chi1minusAlpha2.</p>
<p>A <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&alpha;</mi></mrow></mfenced><mn>100</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>%</mi></mrow></mstyle></math>&nbsp;confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>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><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></mstyle></math>.&nbsp; Therefore, the <em>lower limit</em> of the 90% confidence interval 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><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$s2</mi></mrow><mrow><mi>$ChiAlpha2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Lower</mi></mrow></mstyle></math>, and the <em>upper limit</em> of the 90% confidence interval is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$s2</mi></mrow><mrow><mi>$Chi1minusAlpha2</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$Upper</mi></mrow></mstyle></math>.&nbsp; Therefore, the 90% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>&nbsp;is <em>($Lower, $Upper)</em>.</p>
<p>Using this interval, we can find the 90% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi></mrow></mstyle></math>&nbsp;to be <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>$Lower</mi></mrow></msqrt><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><mi>$Upper</mi></mrow></msqrt></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>$LowerSD</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$UpperSD</mi></mrow></mfenced></mrow></mstyle></math></p>@
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qu.1.7.algorithm=$Data=[rand(1.5,3.5,3),rand(1.5,3.5,3),rand(1.5,3.5,3),rand(1.5,3.5,3),rand(1.5,3.5,3),rand(1.5,3.5,3)];
$ChiValues=maple("
S1:=Statistics[Variance]($Data):
UL:=Statistics[Quantile](ChiSquare(5), 0.05):
LL:=Statistics[Quantile](ChiSquare(5), 0.95):
UL, LL, S1
");
$Chi1minusAlpha2=switch(0, $ChiValues);
$ChiAlpha2=switch(1, $ChiValues);
$s2=switch(2, $ChiValues);
$s=sqrt($s2);
$Upper=(6-1)*$s2/$Chi1minusAlpha2;
$Lower=(6-1)*$s2/$ChiAlpha2;
$UpperSD=sqrt($Upper);
$LowerSD=sqrt($Lower);@
qu.1.7.uid=e9791d2b-cd0a-46ab-82c5-1a36b91bd6bc@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Variance, Confidence Intervals;
  Author=Lorna Deeth;
  Difficulty=Hard;
  Features=None;
  Type=Calculation;
@
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qu.1.7.numbering=alpha@
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qu.1.7.part.2.answer=($LowerSD?0.01, $UpperSD?0.01)@
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qu.1.7.question=<p>The following values are obtained from a random sample that was drawn from a normally distributed population.</p><p>&nbsp;</p><p align="center">$Data</p><p>If we are interested in creating a 90% confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi></mrow></mstyle></math>, the population standard deviation, then:</p><p>&nbsp;</p><p><span>a)&nbsp;&nbsp;Calculate <em>s</em>, the sample standard deviation.</span></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><span>b)&nbsp; What are the values for the lower and upper limits of the confidence interval for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi></mrow></mstyle></math>?</span></span></span></p><p>&nbsp;</p><p><span><span><span>Enter your response in the format: <strong>(Lower limit, Upper limit) </strong>.</span></span></span></p><p>&nbsp;</p><p><span><span><span>Round each of your&nbsp;values&nbsp;to at least 3 decimal places before entering them.</span></span></span></p><p><span><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></span></p>@

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qu.1.8.name=Calculate sample variance, test statistic, conclusion for one-sided hypothesis test (1)@
qu.1.8.comment=<p>a)&nbsp; In order to calculate the test statistic, we first must calculate the sample variance.&nbsp; This can be done using the formula:&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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi mathcolor='#800080'>i</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><msub><mi>x</mi><mrow><mi>i</mi></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mover><mrow><mi>x</mi></mrow><mi>&macr;</mi></mover></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Substituting in the appropriate values, we 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><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$s2</mi></mrow></mstyle></math>.&nbsp; The formula for the test statistic is given as <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>s</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><msubsup><mrow><mi>&sigma;</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></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><msubsup><mrow><mi>&sigma;</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math>&nbsp;is the hypothesized population variance.&nbsp; Substituting in the appropriate values, the test statistic calculates 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><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mn>5</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>$s2</mi></mrow><mrow><mn>60</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$ChiTest</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The alternative hypothesis indicates that we are conducting a one-sided, upper-tailed&nbsp;test.&nbsp; Therefore, to find the p-value, we need to calculate the area to the right of the test statistic, under the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math>distribution&nbsp;with 4 degrees of freedom.&nbsp; Using computer software, we can find the exact&nbsp;p-value to be <em>$pvalue</em>.&nbsp;</p>
<p>&nbsp;</p>
<p>c)&nbsp; Since the p-value is&nbsp;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='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>0.10</mn></mrow></mstyle></math>, there is&nbsp;significant evidence against the null hypothesis.&nbsp; Therefore the&nbsp;null hypothesis is rejected at the 10% level of significance, in favour of the alternative hypothesis that the population variance is greater than 60.&nbsp;&nbsp;</p>@
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qu.1.8.algorithm=$Data=[rand(5,10,4),rand(35,40,4),rand(20,25,4),rand(25,30,4),rand(15,20,4)];
$Values=maple("
S1:=Statistics[Variance]($Data):
CHISQ:=(5-1)*(S1)/60;
X:=Statistics[CDF](ChiSquare(4), CHISQ):
S1, CHISQ, X
");
$s2=switch(0, $Values);
$ChiTest=switch(1, $Values);
$LowerTail=switch(2, $Values);
$pvalue=1-$LowerTail;
condition:gt($pvalue,0.05);
condition:lt($pvalue,0.10);@
qu.1.8.uid=67aa0fe2-d042-49d5-bcdf-17ca14a054cc@
qu.1.8.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Variance, Hypothesis Testing;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Calculation;
@
qu.1.8.weighting=1,1,1@
qu.1.8.numbering=alpha@
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qu.1.8.part.1.answer.num=$ChiTest@
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qu.1.8.part.2.choice.5=p-value < 0.01@
qu.1.8.part.2.fixed=@
qu.1.8.part.2.choice.4=0.01 < p-value < 0.025@
qu.1.8.part.2.question=null@
qu.1.8.part.2.choice.3=0.025 < p-value < 0.05@
qu.1.8.part.2.choice.2=0.05 < p-value < 0.10@
qu.1.8.part.2.choice.1=p-value > 0.10@
qu.1.8.part.2.mode=Non Permuting Multiple Choice@
qu.1.8.part.2.display=vertical@
qu.1.8.part.2.answer=2@
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qu.1.8.part.3.editing=useHTML@
qu.1.8.part.3.fixed=@
qu.1.8.part.3.question=null@
qu.1.8.part.3.choice.2=There is insufficient evidence to reject the null hypothesis, and therefore no significant evidence that the population variance is not 60.@
qu.1.8.part.3.choice.1=There is sufficient evidence to reject the null hypothesis, in favour of the alternative hypothesis that the population variance is&nbsp;greater than&nbsp;60.@
qu.1.8.part.3.mode=Multiple Choice@
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qu.1.8.part.3.answer=1@
qu.1.8.question=<p>The following observations are obtained when a random sample is drawn from a normally distributed population:</p><p>&nbsp;</p><p align="center">$Data</p><p>&nbsp;</p><p>Use this information to test the null hypothesis&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mn>0</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>60</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>&nbsp;against the alternative hypothesis <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>H</mi><mrow><mi>A</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>60</mn></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p><p>&nbsp;</p><p><span>a)&nbsp; What is the value of the test statistic?</span></p><p>&nbsp;</p><p><span>Round your response to at least&nbsp;3 decimal places.</span></p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span>b)&nbsp; The p-value falls within which one of the following ranges:</span></span></span></p><p>&nbsp;</p><p><span><span><span><span>&nbsp;</span><2><span>&nbsp;</span></span></span></span></p><p>&nbsp;</p><p>&nbsp;</p><p><span><span><span><span>c)&nbsp; What conclusion can be made, at the 10% level of significance?</span></span></span></span></p><p>&nbsp;</p><p><span><span><span><span><span>&nbsp;</span><3><span>&nbsp;</span></span></span></span></span></p>@

qu.1.9.mode=Multiple Selection@
qu.1.9.name=Definitions 2: Inference for a Single Population Variance@
qu.1.9.comment=@
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qu.1.9.solution=@
qu.1.9.algorithm=@
qu.1.9.uid=e478bbb1-b825-4997-8017-0b395f87de00@
qu.1.9.info=  Course=Introductory Statistics;
  Topic=Inference for Single Population Variance;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.9.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.1.9.answer=1, 2@
qu.1.9.choice.1=The confidence interval for the population standard deviation is the square root of the confidence interval for the population variance.@
qu.1.9.choice.2=A confidence interval for the population variance can only contain values greater than 0.@
qu.1.9.choice.3=The <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math> value is less than the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mrow><mi>&chi;</mi></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&alpha;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mstyle></math> value.@
qu.1.9.choice.4=Because the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>&chi;</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> distribution is asymmetric, it is not possible to conduct two-sided hypothesis tests.@
qu.1.9.fixed=@

