\documentclass[11pt]{article}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\C{\mathbb{C}}
\usepackage{latexsym,amsmath,amssymb}
\usepackage[all]{xy}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.0in}
\begin{document}
\centerline{MONT 107N -- Understanding Randomness}
\centerline{Final Examination}
\centerline{May 11, 2010}   
\vskip 10pt
\noindent
{\it Directions:}  Do all work in the blue exam booklet.  You must show all work
necessary to justify your answers.  There are 200 possible points, distributed
as indicated in the questions.
\vskip 10pt
\begin{enumerate}                                      
\item Short Answer.  
\begin{enumerate}
\item (10) How are the EV and SE for the sum of $n$ draws with replacement from a box computed?  
\item (10) What does the Central Limit Theorem say?
\item (10) What is a $99\%$ confidence interval for an average?
\item (10) What is the $p$-value of a test of significance? 
\item (10) When is a $t$-test for an average used instead of a $z$-test?  
\end{enumerate}
\item A statistician tosses a coin $100$ times.  His null hypothesis
is that the coin is fair, and his alternative is that 
the coin is biased in favor of heads.  The coin comes up heads $60$ times
 For each of the following, say whether the statement is true or false, and 
explain. 
\begin{enumerate}
\item (10)  If the coin was fair, there would be about a $3\%$ chance of getting $60$ or more
heads in $100$ tosses.
\item (10) Given that $60$ heads are observed, there is only about a $3\%$ chance
that the coin is fair.
\end{enumerate}
\item A gambler is playing roulette. On each play, he bets $\$1$ on a ``split'' (two adjacent
numbers), which pays 17 to 1. That is, if one of his 2 numbers comes up, 
he will win $\$17$ on a $\$1$ bet. Otherwise, he loses the $\$1$.
\begin{enumerate}
\item (5) Construct a box model for the payoff on this bet. 
(Hint: Recall there are a total of 38 slots on a roulette wheel.)
\item (10) Compute the average and SD of the box.
\item (10) If the bet is played 100 times, compute the EV and the SE for the payoff.
\item (10) What are the gambler's chances of coming out ahead on 100 spins of the wheel?
\end{enumerate}
\item  An airline does a market research survey on travel patterns.  It takes a simple
random sample of $225$ people and works out the $95\%$ confidence interval for
the average distance they traveled on vacation in the previous year.  The computed interval
extends from 488 to 592 miles.   Say whether each statement below is true or false
and give reasons.  If there is not enough information given to decide, explain what
other information you would need.
\begin{enumerate}
\item (5) The average distance traveled was about 540 miles.
\item (5) The SD of the distances traveled was about 200 miles.
\item (5) The probability histogram for averages of samples of size 225
drawn from the population is probably close to (scaled, shifted) normal curve.
\item (5) The probability histogram for the distances traveled by randomly chosen
individuals in the population is close to a (scaled, shifted) normal curve.
\item (5) If the sample size is increased to $450$, the width of the 
confidence interval will always be about half as wide.   
\end{enumerate}
\item A simple random sample of 1000 persons is taken to determine the 
percentage of Democrats in a large population.  It turns out that
$543$ of the people in the sample are Democrats.
\begin{enumerate}
\item (10)  Compute a $95\%$ confidence interval for the percentage
of Democrats in the population.
\item (10)  True or False and explain:  There is about a $5\%$ chance that
the actual percentage of Democrats is outside the interval found in part A.  
\end{enumerate}
\item The National Assessment of Educational Progress tested 17-year-olds on mathematics
in 1990 and again in 2004.  In 1990, the average score was 305 and the SD was 34.  In
2004, the average score was 307 and the SD was 27.  In both years, simple random
samples of size 1000 were used.  Can the difference be explained as chance variation, 
or have average mathematics scores really gone up?
\begin{enumerate}
\item (5) Should you use a one-sample or a two-sample $z$-test for this kind of question?
\item (10) Formulate null and alternative hypotheses in terms of a box model.  (Say how 
many boxes there are, how many tickets go in each, and what is on the tickets -- scores
or 0/1's.) 
\item (10) Carry out the test, find the associated $p$-value, and discuss your results.
\end{enumerate}
\item A New York Times/CBS poll conducted between April 5 and April 12, 2010 
included the question ``Do you approve or disapprove of the way your Representative in Congress is
handling his or her job?''  The poll was carried out in two stages.  First a large national 
simple random sample was asked the question.  The results were that
$46\%$ of respondents said they approved, $36\%$ said they disapproved, and
$18\%$ said they had no opinion. 
Then the same question was asked of a separate sample of 881 people, 
all of whom had identified themselves as supporters of the ``Tea Party'' 
movement.  The responses broke down like this:
\[
\begin{array}{cccccccc}\hline
         & \text{Approve} &  \text{Disapprove}  &  \text{No opinion} & \text{Total}\\ \hline
\text{``Tea Party'' Sample} &   352   &   432   &    97   &    881\\
\end{array}
\]
(Data is adapted from the New York Times web site.)  It seems from the numbers that 
``Tea Party'' supporters may differ from the general population when it comes to
their opinions concerning their Congressional Representatives.  
But is this a real difference, or could it just be a product of chance
variation in the sampling process?   
\begin{enumerate}
\item (5) Formulate a null and alternative hypothesis for this question.
\item (5) What significance test should you use to decide between these?
\item (10) Carry out your test of signficance. What do you get for a test statistic and observed
          significance level ($p$-value)?
\item (5) Does this data give evidence of a real difference between ``Tea Party''
supporters and the public as a whole?  Explain.  
\end{enumerate}
\end{enumerate}
\centerline{\it Have a safe, enjoyable, and productive summer!}
\end{document}
