qu.1.topic=Sampling Distributions@

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qu.1.1.name=Sampling Distribution of the Sample Mean: Calculate mean, standard error, normality (3)@
qu.1.1.comment=<p>a)&nbsp; The mean of the sampling distribution of the sample mean, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&mu;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi mathcolor='#800080'>_</mi></mrow></mover></mrow></msub></mrow></mstyle></math>, will be the same as the mean of the population from which the sample was taken.&nbsp; 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>&mu;</mi><msub><mi></mi><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The standard deviation of the sampling distribution of the sample mean, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub></mrow></mstyle></math>, is&nbsp;given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mi>&sigma;</mi></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi mathcolor='#800080'>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sigmaXbar</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp;&nbsp;Even though the sample size is small, it is stated that&nbsp;the population from which we are sampling is normally distributed.&nbsp; Therefore, we can conclude&nbsp;that the sampling distribution of the sample mean is approximately normal.</p>@
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  Topic=Sampling Distribution of the Sample Mean;
  Author=Lorna Deeth;
  Difficulty=Easy;
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  Type=Calculation;
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qu.1.1.question=<p>A&nbsp;random sample of size $n was taken from a normally distributed&nbsp;population with a population&nbsp;mean $mean and a population&nbsp;standard deviation $sigma.</p><p>&nbsp;</p><p>Determine each of the following about the sampling distribution of the sample mean.</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places where appropriate.</p><p>&nbsp;</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><msub><mrow><mi>&mu;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi mathcolor='#800080'>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>c)&nbsp;&nbsp;Can we conclude that&nbsp;the sampling distribution of the sample mean is&nbsp;approximately normal?&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></p>@

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qu.1.2.name=Sampling Distribution of the Sample Mean: Calculate mean, standard error, normality (1)@
qu.1.2.comment=<p>a)&nbsp; The mean of the sampling distribution of the sample mean, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&mu;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi mathcolor='#800080'>_</mi></mrow></mover></mrow></msub></mrow></mstyle></math>, will be the same as the mean of the population from which the sample was taken.&nbsp; 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>&mu;</mi><msub><mi></mi><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The standard deviation of the sampling distribution of the sample mean, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub></mrow></mstyle></math>, is&nbsp;given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mi>&sigma;</mi></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi mathcolor='#800080'>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sigmaXbar</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; There is nothing to indicate whether or not the population from which we are sampling&nbsp;is normally distributed, and we cannot rely on the Central Limit Theorem because the sample size is small.&nbsp; Therefore, we cannot determine whether or not the sampling distribution is approximately normal.</p>@
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qu.1.2.question=<p>A&nbsp;random sample of size $n was taken from a&nbsp;population with a population mean $mean and a population standard deviation $sigma.</p><p>&nbsp;</p><p>Determine each of the following about the sampling distribution of the sample mean.</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places where appropriate.</p><p>&nbsp;</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><msub><mrow><mi>&mu;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi mathcolor='#800080'>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>c)&nbsp;&nbsp;Can we conclude that&nbsp;the sampling distribution of the sample mean is&nbsp;approximately normal?&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></p>@

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qu.1.3.info=  Course=Introductory Statistics;
  Topic=Sampling Distribution of the Sample Proportion;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Concept;
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qu.1.3.part.1.choice.4=As the sample size increases, the sampling distribution of the population proportion becomes more normal.@
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qu.1.3.part.1.choice.3=If the population proportion is close to either 0 or 1, then only a small sample needs to be taken in order for the sampling distribution of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>to be approximately normal.@
qu.1.3.part.1.choice.2=If the population proportion is close to 0.5, then the sampling distribution of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>is approximately normal.@
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qu.1.4.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mi>p</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>q</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>n</mi></mrow></msqrt></mrow></mstyle></math>@
qu.1.4.part.1.question=null@
qu.1.4.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>pq</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></mstyle></math>@
qu.1.4.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>pq</mi></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>@
qu.1.4.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'></mi></mrow><mrow><msqrt><mrow><mfrac><mrow><mi>pq</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msqrt></mrow></mstyle></math>@
qu.1.4.part.1.mode=Multiple Choice@
qu.1.4.part.1.display=vertical@
qu.1.4.part.1.answer=1@
qu.1.4.question=<p>The standard error of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mover><mrow><mi>p</mi></mrow><mi>&#x005e;</mi></mover></mrow></mstyle></math>is given by which one of the following formulas?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.5.mode=Multiple Selection@
qu.1.5.name=Definitions 2: Sampling Distribution of the mean@
qu.1.5.comment=@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=@
qu.1.5.uid=d1f32d5e-1bca-47e7-a1e0-fa1b609d0fe0@
qu.1.5.info=  Course=Introductory Statistics;
  Topic=Sampling Distribution of the mean;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.5.question=<p>Which of the following statements are TRUE?</p>
<p>&nbsp;</p>
<p>Note that there may be more than one correct answer; select all that are true.</p>@
qu.1.5.answer=1, 2@
qu.1.5.choice.1=Statistics have sampling distributions.@
qu.1.5.choice.2=The value of a parameter does not vary from sample to sample.@
qu.1.5.choice.3=The value of a statistic does not vary from sample to sample.@
qu.1.5.choice.4=All else being equal, the standard deviation of the sampling distribution of the sample mean will be smaller for n = 10 than for n = 40.@
qu.1.5.choice.5=The sampling distribution of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi></mrow></mstyle></math> is always approximately normal for n > 30.@
qu.1.5.fixed=@

qu.1.6.mode=Inline@
qu.1.6.name=Definition of the Central Limit Theorem@
qu.1.6.comment=@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=@
qu.1.6.uid=bda1ea1b-2b80-4dcb-a7cc-c7a569ac148f@
qu.1.6.info=  Course=Introductory Statistics;
  Topic=Sampling Distributions, Central Limit Theorem;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Concept;
@
qu.1.6.weighting=1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.choice.5=If the distribution of the population is nonnormal, it can be normalized by taking a large sample size.@
qu.1.6.part.1.fixed=@
qu.1.6.part.1.choice.4=For nonnormally distributed populations, the sampling distribution of the sample mean will be approximately normal, regardless of the sample size.@
qu.1.6.part.1.question=null@
qu.1.6.part.1.choice.3=For large sample sizes, the sampling&nbsp;distribution of the population mean is approximately normal, regardless of the distribution of the population.@
qu.1.6.part.1.choice.2=In large populations, the distribution of the population mean is approximately normal.@
qu.1.6.part.1.choice.1=The sampling distribution of the sample mean is approximately normal for large sample sizes, regardless of the distribution of the population.@
qu.1.6.part.1.mode=Multiple Choice@
qu.1.6.part.1.display=vertical@
qu.1.6.part.1.answer=1@
qu.1.6.question=<p>Which one&nbsp;of the following statements is the best definition of the Central Limit Theorem?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.7.mode=Inline@
qu.1.7.name=Definition of a biased estimator@
qu.1.7.comment=<p>In order to be an unbiased estimator, the expected value of the estimator must equal the true value.&nbsp; Therefore, a biased estimator is one in which the expected value of the estimator does not equal the true value.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=@
qu.1.7.uid=39a0f495-e677-4bb8-b57a-8d2bd02be883@
qu.1.7.info=  Course=Introductory Statistics;
  Topic=Sampling Distributions, Biased Estimator;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Concept;
@
qu.1.7.weighting=1@
qu.1.7.numbering=alpha@
qu.1.7.part.1.name=sro_id_1@
qu.1.7.part.1.editing=useHTML@
qu.1.7.part.1.choice.5=An estimator that is not normally distributed.@
qu.1.7.part.1.fixed=@
qu.1.7.part.1.choice.4=An estimator that has two possible values.@
qu.1.7.part.1.question=null@
qu.1.7.part.1.choice.3=An estimator&nbsp;whose value&nbsp;is not exactly equal to the true value.@
qu.1.7.part.1.choice.2=An estimator that has a very large standard deviation.@
qu.1.7.part.1.choice.1=An estimator whose expected value does not equal the true value.@
qu.1.7.part.1.mode=Multiple Choice@
qu.1.7.part.1.display=vertical@
qu.1.7.part.1.answer=1@
qu.1.7.question=<p>Which of the following statements is the best definition of a <strong>biased</strong> estimator?</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.8.mode=Inline@
qu.1.8.name=Formula for the standard deviation of the sampling distribution of the sample mean@
qu.1.8.comment=@
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qu.1.8.solution=@
qu.1.8.algorithm=@
qu.1.8.uid=6ac6e628-dbea-4fd6-8f88-954d8dfe5be2@
qu.1.8.info=  Course=Introductory Statistics;
  Topic=Sampling Distribution of the Sample Mean;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Concept;
@
qu.1.8.weighting=1@
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qu.1.8.part.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>&mu;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>@
qu.1.8.part.1.fixed=@
qu.1.8.part.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mfenced open='(' close=')' separators=','><mrow><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&sigma;</mi></mrow></mfenced><mrow><mi>n</mi></mrow></mfrac></mrow></mstyle></math>@
qu.1.8.part.1.question=null@
qu.1.8.part.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>&sigma;</mi><mrow><mi>n</mi></mrow></mfrac></mrow></mstyle></math>@
qu.1.8.part.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&sigma;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mrow></mstyle></math>@
qu.1.8.part.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi>&sigma;</mi><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mstyle></math>@
qu.1.8.part.1.mode=Multiple Choice@
qu.1.8.part.1.display=vertical@
qu.1.8.part.1.answer=1@
qu.1.8.question=<p>If a sample of size <em>n</em> is drawn from a population with mean <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>&mu;</mi></mrow></mstyle></math>&nbsp;and standard deviation <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>, then the standard deviation of the sampling distribution of the sample mean is given by the formula:</p><p>&nbsp;</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.9.mode=Multiple Selection@
qu.1.9.name=Definitions 1: Sampling Distribution of the mean@
qu.1.9.comment=@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=@
qu.1.9.uid=db39936b-7479-4a74-b4be-e0fa4afbce03@
qu.1.9.info=  Course=Introductory Statistics;
  Topic=Sampling Distribution of the mean;
  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=4, 5@
qu.1.9.choice.1=If the sampling distribution of the sample mean is approximately normal, then the population from which the samples were drawn must have been normally distributed.@
qu.1.9.choice.2=The standard deviation of the sampling distribution of the sample mean depends on the value of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&mu;</mi></mrow></mstyle></math>.@
qu.1.9.choice.3=We cannot possibly determine any characteristics of a statistic's sampling distribution without repeatedly sampling from the population.@
qu.1.9.choice.4=The sampling distribution of the sample mean is approximately normal for large sample sizes, and is sometimes approximately normal for small sample sizes.@
qu.1.9.choice.5=If we quadruple the sample size, the standard deviation of the sampling distribution of the sample mean would decrease by a factor of 2.@
qu.1.9.fixed=@

qu.1.10.mode=Inline@
qu.1.10.name=Sampling Distribution of the Sample Mean: Calculate mean, standard error, normality (2)@
qu.1.10.comment=<p>a)&nbsp; The mean of the sampling distribution of the sample mean, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&mu;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi mathcolor='#800080'>_</mi></mrow></mover></mrow></msub></mrow></mstyle></math>, will be the same as the mean of the population from which the sample was taken.&nbsp; 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>&mu;</mi><msub><mi></mi><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&mu;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$mean</mi></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>b)&nbsp; The standard deviation of the sampling distribution of the sample mean, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub></mrow></mstyle></math>, is&nbsp;given by the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mi>&sigma;</mi></mrow><mrow><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></mfrac></mrow></mrow></mstyle></math>.&nbsp; Therefore, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi mathcolor='#800080'>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>$sigma</mi><mrow><msqrt><mrow><mi>$n</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$sigmaXbar</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>.</p>
<p>&nbsp;</p>
<p>c)&nbsp; There is nothing to indicate whether or not the population from which we are sampling is normally distributed, however we can apply the Central Limit Theorem because the sample size is large.&nbsp; Therefore, we can conclude&nbsp;that the sampling distribution of the sample mean is approximately normal.</p>@
qu.1.10.editing=useHTML@
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qu.1.10.algorithm=$n=range(100,150);
$mean=range(25,30);
$sigma=range(4,8);
$sigmaXbar=$sigma/sqrt($n);@
qu.1.10.uid=320f366e-2c0b-4ea2-bd93-05e875cd1106@
qu.1.10.info=  Course=Introductory Statistics;
  Topic=Sampling Distribution of the Sample Mean;
  Author=Lorna Deeth;
  Difficulty=Easy;
  Features=None;
  Type=Calculation;
@
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qu.1.10.part.3.credit.2=0.0@
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qu.1.10.question=<p>A&nbsp;random sample of size $n was taken from a&nbsp;population with a population&nbsp;mean $mean and a population standard deviation $sigma.</p><p>&nbsp;</p><p>Determine each of the following about the sampling distribution of the sample mean.</p><p>&nbsp;</p><p>Round your answer to at least&nbsp;3 decimal places where appropriate.</p><p>&nbsp;</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><msub><mrow><mi>&mu;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mi>&sigma;</mi></mrow><mrow><mover><mrow><mi mathcolor='#800080'>x</mi></mrow><mrow><mi mathcolor='#800080'>_</mi></mrow></mover></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></p><p>&nbsp;</p><p>c)&nbsp;&nbsp;Can we conclude that&nbsp;the sampling distribution of the sample mean is&nbsp;approximately normal?&nbsp;&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span></p>@

qu.1.11.mode=Inline@
qu.1.11.name=Definitions 1&2: Random Selection of T/F@
qu.1.11.comment=@
qu.1.11.editing=useHTML@
qu.1.11.solution=@
qu.1.11.algorithm=$k1=rint(10);
$k2=rint(10);
$k3=rint(10);
$k4=rint(10);
$k5=rint(10);
$z=maple("S := $k1,$k2,$k3,$k4,$k5:
floor( nops({S})/nops([S]) )");
condition: $z;
$a=("'Statistics have sampling distributions.'");
$b=("'The value of a parameter does not vary from sample to sample.'");
$c=("'The value of a statistic does not vary from sample to sample.'");
$d=("'All else being equal, the standard deviation of the sampling distribution of the sample mean will be smaller for n = 10 than for n = 40.'");
$e=("'If the sampling distribution of the sample mean is approximately normal, then the population from which the samples were drawn must have been normally distributed.'");
$f=("'We cannot possibly determine any characteristics of a statistic�s sampling distribution without repeatedly sampling from the population.'");
$g=("'The sampling distribution of the sample mean is approximately normal for large sample sizes, and is sometimes approximately normal for small sample sizes.'");
$h=("'If we quadruple the sample size, the standard deviation of the sampling distribution of the sample mean would decrease by a factor of 2.'");
$ij=maple("
i1:=convert(cat(`The sampling distribution of `, MathML[ExportPresentation](mu), ` is always approximately normal for n > 30.`),string):
j1:=convert(cat(`The standard deviation of the sampling distribution of the sample mean depends on the value of `, MathML[ExportPresentation](mu), `.`),string):
i1, j1
");
$i=switch(0, $ij);
$j=switch(1, $ij);
$Answers=["'True'","'True'","'False'","'False'","'False'","'False'","'True'","'True'","'False'","'False'"];
$Distractors=["'False'","'False'","'True'","'True'","'True'","'True'","'False'","'False'","'True'","'True'"];
$Q1=switch($k1, $a,$b,$c,$d,$e,$f,$h,"$i","$j");
$A1=switch($k1, $Answers);
$D1=switch($k1, $Distractors);
$Q2=switch($k2, $a,$b,$c,$d,$e,$f,$h,"$i","$j");
$A2=switch($k2, $Answers);
$D2=switch($k2, $Distractors);
$Q3=switch($k3, $a,$b,$c,$d,$e,$f,$h,"$i","$j");
$A3=switch($k3, $Answers);
$D3=switch($k3, $Distractors);
$Q4=switch($k4, $a,$b,$c,$d,$e,$f,$h,"$i","$j");
$A4=switch($k4, $Answers);
$D4=switch($k4, $Distractors);
$Q5=switch($k5, $a,$b,$c,$d,$e,$f,$h,"$i","$j");
$A5=switch($k5, $Answers);
$D5=switch($k5, $Distractors);@
qu.1.11.uid=6ba1b4e3-6e07-497f-8000-65a416ae1399@
qu.1.11.info=  Course=Introductory Statistics;
  Topic=Sampling Distribution of the mean;
  Author=Lorna Deeth;
  Difficulty=Medium;
  Features=None;
  Type=Concept;
@
qu.1.11.weighting=1,1,1,1,1@
qu.1.11.numbering=alpha@
qu.1.11.part.1.grader=exact@
qu.1.11.part.1.name=sro_id_1@
qu.1.11.part.1.editing=useHTML@
qu.1.11.part.1.display.permute=true@
qu.1.11.part.1.question=(Unset)@
qu.1.11.part.1.answer.2=$D1@
qu.1.11.part.1.answer.1=$A1@
qu.1.11.part.1.mode=List@
qu.1.11.part.1.display=menu@
qu.1.11.part.1.credit.2=0.0@
qu.1.11.part.1.credit.1=1.0@
qu.1.11.part.2.grader=exact@
qu.1.11.part.2.name=sro_id_2@
qu.1.11.part.2.editing=useHTML@
qu.1.11.part.2.display.permute=true@
qu.1.11.part.2.question=(Unset)@
qu.1.11.part.2.answer.2=$D2@
qu.1.11.part.2.answer.1=$A2@
qu.1.11.part.2.mode=List@
qu.1.11.part.2.display=menu@
qu.1.11.part.2.credit.2=0.0@
qu.1.11.part.2.credit.1=1.0@
qu.1.11.part.3.grader=exact@
qu.1.11.part.3.name=sro_id_3@
qu.1.11.part.3.editing=useHTML@
qu.1.11.part.3.display.permute=true@
qu.1.11.part.3.question=(Unset)@
qu.1.11.part.3.answer.2=$D3@
qu.1.11.part.3.answer.1=$A3@
qu.1.11.part.3.mode=List@
qu.1.11.part.3.display=menu@
qu.1.11.part.3.credit.2=0.0@
qu.1.11.part.3.credit.1=1.0@
qu.1.11.part.4.grader=exact@
qu.1.11.part.4.name=sro_id_4@
qu.1.11.part.4.editing=useHTML@
qu.1.11.part.4.display.permute=true@
qu.1.11.part.4.question=(Unset)@
qu.1.11.part.4.answer.2=$D4@
qu.1.11.part.4.answer.1=$A4@
qu.1.11.part.4.mode=List@
qu.1.11.part.4.display=menu@
qu.1.11.part.4.credit.2=0.0@
qu.1.11.part.4.credit.1=1.0@
qu.1.11.part.5.grader=exact@
qu.1.11.part.5.name=sro_id_5@
qu.1.11.part.5.editing=useHTML@
qu.1.11.part.5.display.permute=true@
qu.1.11.part.5.question=(Unset)@
qu.1.11.part.5.answer.2=$D5@
qu.1.11.part.5.answer.1=$A5@
qu.1.11.part.5.mode=List@
qu.1.11.part.5.display=menu@
qu.1.11.part.5.credit.2=0.0@
qu.1.11.part.5.credit.1=1.0@
qu.1.11.question=<p>Determine whether the following statements are TRUE or FALSE:</p><p>a)&nbsp;<span>&nbsp;</span><1><span>&nbsp;</span> $Q1</p><p>b)&nbsp;<span>&nbsp;</span><2><span>&nbsp;</span> $Q2</p><p>c)&nbsp;<span>&nbsp;</span><3><span>&nbsp;</span> $Q3</p><p>d)&nbsp;<span>&nbsp;</span><4><span>&nbsp;</span> $Q4</p><p>e)&nbsp;<span>&nbsp;</span><5><span>&nbsp;</span> $Q5</p>@

