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\centerline{MONT 107N -- Understanding Randomness}
\centerline{Some Sample Final Exam Questions}
\centerline{April 29, 2010}   
\begin{enumerate}                                      
\item Briefly explain the following chance and statistical concepts.
\begin{enumerate}
\item Central Limit Theorem.
\item Law of Averages.
\item $95\%$ conﬁdence interval.
\item $\chi^2$ statistic.
\item Null hypothesis of a test of significance.
\item $p$-value of a test of significance.
\end{enumerate}
\item Short Answer:
\begin{enumerate}
\item (Hypothetical) A college has an elective quantitative reasoning course
            for first-year students. Each year
            approximately one fifth of the first-year students elect to take this course. The college
            does a study of the grades of first-year students. The study shows that after the first-
            year of college, the students who elect to take this math course have an average GPA
            for their other three courses that is .1 units higher than the average GPA for the other
            students for all their courses.  Based on this data, the Mathematics department argues that
            this course raises student GPAs and that every student should be required to take it.
            Does the Mathematics Department have a good argument or is it possibly flawed? If it is a good
            argument, explain why, if not, how would you correct the study?
\item Why are we justified in using a normal curve to make estimates about draws from box
            models regardless of whether or not the data in the box is normally distributed?
\item In major national polls, which guide politicians and political candidates in their decision
            making and play a prominent role in the national media, the sample size is usually 1000
            or so. Why are samples of this size used? In particular, what are the benefits of using
            a sample size of 1000 as opposed to smaller sample sizes or larger sample sizes?
\item If you want to get polling results with comparable accuracy, do you need a bigger
sample size if the polling is done in California than you do if the polling is done
in Rhode Island?  Explain.
\item True or False and explain:  If the $p$-value of a test of significance works
out to $p = .05$ then we can say that the null hypothesis was false.  
\item True or False and explain:  Increasing the sample size will always
decrease the width of a $95\%$ confidence interval for an average.  
\end{enumerate}
\item A gambler is playing roulette. On each play, he bets $\$1$ on four adjacent numbers, which
   pays 8 to 1. That is, if one of his 4 numbers comes up, he will win $\$8$ on a $\$1$ bet.
   Otherwise, he loses $\$1$.
\begin{enumerate}
\item Construct a box model for this bet. (Hint: Recall there are a total of 38 slots on a
        roulette wheel.)
\item Compute the average and SD of the box.
\item If the bet is played 50 times, compute the EV and the SE for the bet.
\item What are the gambler's chances of winning $\$40$ or more on 50 spins of the wheel?
\end{enumerate}
\item A CBS News poll released on November 24, 2008 surveyed a random sample of 706 adults nationwide.
It reported “that $61\%$ of Americans say it is too soon for movies to be made about the current
Iraq war.”
\begin{enumerate}
\item What box model would you use to represent this poll?
\item Use your model to determine a $95\%$ confidence interval for the percent of Americans
who actually feel this way.
\item True or False and explain:  There is a $95\%$ chance that the actual percent of 
Americans who feel this way is in the interval you computed in part b.  
\end{enumerate}
\item Before a person gives blood, the Red Cross requires that the hemoglobin in their blood mea-
   sures 12.0 or higher on an electronic scan.  There is typically some error in the measurements.
   (The following is hypothetical.) Suppose that such a device is being calibrated by five read-
   ings on a standardized sample known to have a hemoglobin content of 12.0. The five readings
   are 11.5, 11.9, 12.0, 12.1, 12.1 for an average of 11.92. Based on these readings, we want to
   do a test of significance to determine whether the device is calibrated correctly or not.
\begin{enumerate}
\item What box model would you use to model this calibration process?
\item Formulate a null and alternative hypothesis for the difference between 12.0 and 11.92.
\item What significance test should you use? Why?
\item Carry out your test of significance. What do you get for a a test statistic and observed
      significance level?
\item Is the device calibrated correctly, or is it biased?
\end{enumerate}
\item (This problem is adapted from ``Heart rate in yoga asana practice: A comparison of styles,''
   Journal of Bodywork and Movement Therapies Volume 11, Issue 1, January 2007, Pages
   91-95.)
   “Yoga is often recommended for stress relief, yet some of the more fitness-oriented styles
   of yoga can be vigorous forms of exercise. The purpose of this study was to investigate
   differences in heart rate during the physical practice of yoga postures, breathing exercises,
   and relaxation.” The study led groups of participants through three different styles of yoga:
    astanga yoga, hatha yoga, and ``gentle'' yoga. Participants wore heart rate
   monitors during the sessions and their heart rates were monitored repeatedly throughout the
   sessions. Assume there were three independent groups of 50 participants each in the study.
Each group was put through a practice session with each style of yoga. The average and standard deviation
   for the heart rates (in beats per minute) of the participants for each style of yoga are given below:
\[
\begin{array}{cccc}\hline
\text{Yoga style} &  \text{Average heart rate (bpm)} & \text{SD}\\ \hline
\text{astanga}    &          95                   &   12.84\\
\text{hatha}      &          80                   &    9.32\\
\text{``gentle''} &          74                   &    7.41\\
\end{array}
\]
The researchers then applied two-sample $z$-tests to each pair of styles and concluded “that
there may be different fitness benefits for different styles of yoga practice.”
\begin{enumerate}
\item Apply a two-sample $z$-test to the astanga and hatha yoga data.  What is your $p$-value?
\item Apply a two-sample $z$-test to the hatha and gentle yoga data.  What is this $p$-value?
\item Were the researchers' conclusions justified? (Note -- the higher the average
heart rate attained, the higher the fitness benefit.)
\end{enumerate}
\item The December 7, 2007 Gallup Poll repeated a number of questions about how people perceive
    the relative ability of the Democratic Party and the Republican Party to handle ten important
    issues. The poll was of adults 18 years old or older. On the issue of health care, the poll
    asked the following question:
          {\it Do you think the Republican Party or Democratic Party would do a better job of
          dealing with each of the following issues and problems? How about health care
          policy?}
    Here are the responses for more recent poll and the poll of January 2004:
\[
\begin{array}{cccccccc}\hline
       & \text{Republicans} &  \text{Democrats}  &  \text{No difference/no opinion} & \text{Total}\\ \hline
\text{2004 Jan. 9-11} &               357       &          578    &           115   &      1050\\
\text{2007 Nov. 30-Dec. 2} &          291       &          592    &           120   &      1003\\
\end{array}
\]
(Data was adapted from the Gallup web site.) The natural question to ask is whether or not
    this represented a real shift in public opinion.
\begin{enumerate}
\item Formulate a null and alternative hypothesis for this question.
\item What significance test should you use to determine the answer?
\item Carry out your test of signficance. What do you get for a test statistic and observed
          significance level?
\item Did this data represent a real shift in public opinion?
\end{enumerate}
\end{enumerate}
\end{document}
