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\centerline{MONT 101N -- Analyzing Environmental Data}
\centerline{Some Sample Final Exam Questions}
\centerline{April 27, 2018}   
\begin{enumerate}                                      
\item Briefly explain the following chance and statistical concepts.
\begin{enumerate}
\item Central Limit Theorem
\item $95\%$ conﬁdence interval
\item $t$-distribution
\item standard normal distribution
\item Null hypothesis of a test
\item $p$-value of a test of hypotheses
\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 sample means
with sample sizes $N > 30$ even if the population the individual $Y_i$ are
sampled from is not 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 $N$ will \emph{always}
decrease the width of a $95\%$ confidence interval for an average.  (Assume $N > 30$ to start with.)
\end{enumerate}

\item Let $Z$ represent a standard normal random variable and $X$ represent a normal
random variable with $\mu = 4.5$ and $\sigma = 1.3$.  
\begin{enumerate}
\item Find $P(1.24 < Z < 2.0)$
\item Find $P(-0.4 < Z < 1.33)$
\item Find $P(X > 5)$
\item Find $P(4 < X < 5.4)$ 
\item Suppose that values of $X$ are sampled $n = 12$ times independently.
What is the probability that $4$ out of the $12$ values satisfy $4 < X < 5.4$.  
\item Now suppose you have a (very) large urn containing 10000 balls, 
$6069$ of them blue and $3031$ of them red.  Suppose you picked 12 of them 
at random \emph{without replacement}.  What is the probability that you get exactly
$4$ blue balls?
\end{enumerate}

\item A Gallup poll released on April 3, 2017 surveyed a random sample of 706 adults nationwide.
It reported ``that $59\%$ of Americans say the environment should be prioritized over energy 
production.''
\begin{enumerate}
\item Determine a $95\%$ confidence interval for the percent of Americans
who actually feel this way.  Use the ``conservative'' method of computing the standard
error.
\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 hypothesis test to determine whether the device is calibrated correctly or not.
\begin{enumerate}
\item Formulate a null and alternative hypothesis for the difference between 12.0 and 11.92.
\item What  test should you use? Why?
\item Carry out your test. What do you get for 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 one 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}
\end{enumerate}
\end{document}
