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\noindent
{\bf 14.31, Fall 2000 -- Problem Set One (Review)} \\ 
Due: Tuesday, September 19 \\ 
Clearly indicate your answer.  No credit given for hard to find answers. \\ [8pt]

\noindent
1. Suppose that a student is preparing to take the SAT exam.  Why is her
eventual SAT score properly viewed as a random variable? \\ [8pt]

\noindent
2. Suppose that the joint probability distribution, p(x,y), of variables X and
Y is
\begin{center}
\begin{tabular}{r|rrr} \hline
\multicolumn{1}{c|}{} &
\multicolumn{3}{c}{X} \\
\multicolumn{1}{c|}{Y} &
\multicolumn{1}{c}{1} &
\multicolumn{1}{c}{2} &
\multicolumn{1}{c}{3} \\ \hline
1 & 0.03 & 0.06 & 0.06 \\
2 & 0.02 & 0.04 & 0.04 \\
3 & 0.09 & 0.18 & 0.18 \\
4 & 0.06 & 0.12 & 0.12 \\ \hline
\end{tabular}
\end{center}
Determine \\
a) the marginal unconditional probability distribution of X and Y. \\
b) the conditional probability distribution $p(X|Y_{i})$ and $p(Y|X_{i})$. \\
c) the conditional expectation of X and Y, $E(X|Y_{i})$ and $E(Y|X_{i})$. \\ [8pt]

\noindent
3. The joint probability p(x,y) of random variables X and Y where X is the first
year rate of return (\%) expected from project A and Y is the first year rate of
return (\%) expected from project B is given by
\begin{center}
\begin{tabular}{r|rrrr} \hline
\multicolumn{1}{c|}{} &
\multicolumn{4}{c}{X} \\
\multicolumn{1}{c|}{Y} &
\multicolumn{1}{c}{-10} &
\multicolumn{1}{c}{0} &
\multicolumn{1}{c}{20} &
\multicolumn{1}{c}{30} \\ \hline
20 & 0.27 & 0.08 & 0.16 & 0.00 \\
50 & 0.00 & 0.04 & 0.10 & 0.35 \\ \hline
\end{tabular}
\end{center}
a) Calculate the expected rate of return on project A, E(X). \\
b) Calculate the expected rate of return on project B, E(Y). \\
c) Are the rates of return on the two projects independent? \\ [8pt]

\noindent
4. Suppose that the height of men in a population is normally distributed with
mean$=\mu$ inches and standard deviation $\sigma=2.5$ inches.  A sample of
100 men drawn randomly from this population had an average height of 67
inches. \\
a) Establish a 95\% confidence interval for the mean height in the population
as a whole. \\
b) Could this sample with a mean of 67 have come from a population with a
mean value of 69? \\ [8pt]

\noindent
5. Let X denote the prison sentence in years, for people convicted of auto
theft in a particular state in the United States.  Suppose that the pdf of
X is given by 
\begin{eqnarray*}
f(x)=(1/9)x^{2} \, , 0 < x < 3
\end{eqnarray*}
Use integration to find the expected prison sentence.  What is the variance
of the random variable X with this density function? \\ [8pt]

\noindent
6. Suppose that the rainfall in a certain locality is normally distributed with
$E(X)=30.0$ inches.  (This value has been established from a long history of
weather data).  In recent years, certain climatological changes seem to be affecting,
among other things, the annual precipitation.  It is hypothesized that in fact the
annual rainfall has increased.  The variance is assumed to be unknown since the
suggested climatological changes may affect the variability of the rainfall, and
hence past records on the variance are not meaningful.  Assume that the past
eight years have yielded the following annual precipitation (inches): \\
\indent 34.1, 33.7, 27.4, 31.1, 30.9, 35.2, 28.4, 32.1 \\
Use a t-test to test $H_{0}:\mu=30.0$ and $H_{i}:\mu > 30.0$. \\ [8pt]

\noindent
7. You have data on a random sample of 300 college students.  The
students were classified with respect to the size of the high school from which
they graduated and with respect to their freshman year grade point average.
\begin{center}
\begin{tabular}{l|rrr} \hline
\multicolumn{1}{c|}{} &
\multicolumn{3}{c}{High School} \\ 
\multicolumn{1}{c|}{Record} &
\multicolumn{1}{c}{Small} &
\multicolumn{1}{c}{Medium} &
\multicolumn{1}{c}{Large} \\ \hline
Above 2.5 average & 18 & 51 & 46 \\
Below 2.5 average & 42 & 79 & 64 \\ \hline
\end{tabular}
\end{center}
Use a Chi-square test to determine if there is any relationship between
high school size and freshman grade point average. \\ [8pt]

\noindent
8. Suppose that at a large university, college grade point average, GPA,
and SAT score, SAT, are related by the conditional expectation E(GPA|SAT)=0.70+
0.002SAT. \\
i) Find the expected GPA when SAT=800.  Find E(GPA|SAT=1,400).  Comment on the
difference. \\
ii) If the average SAT in the university is 1,100, what is the average GPA?
(Hint: recall that E[E(Y|X)]=E(Y).) \\ [8pt]

\noindent
9. In some years some mutual funds will outperform the market (the return from
holding their shares is higher than the return from holding a portfolio such
as the S\&P 500).  Suppose that you examine the performance of 4,170 funds
over a ten year period.  If performance relative to the market is random then
each fund has a 50-50 change of outperforming the market in any year and its
performance is independent from year to year. \\
i) If performance relative to the market is random, what is the probability
that any particular fund outperforms the market in all 10 years? \\
ii) Find the probability that at least one fund out of 4,170 outperforms the
market in all 10 years.  What do you make of your answer? \\ [8pt]

\noindent
10. For a randomly selected county in the United States, let X represent the
proportion of adults over age 65 who are employed, or the elderly employment
rate.  Then, X is restricted a value between 0 and 1.  Suppose that the
cumulative distribution function for X is given by $F(x)=3x^{2}-2x^{3}$ for
$0 \leq x \leq 1$. find the probability that the elderly employment rate is
at least 0.6 (60\%). \\ [8pt]

\noindent
11. Suppose that the salary new graduates measured in {\em thousands} of dollars
is 32.3, with a standard deviation of 14.6.  What are the mean and standard
deviation when salary is measured in dollars. \\ [8pt]

\noindent
12. Let $Y_{1}$, $Y_{2}$, $Y_{3}$, and $Y_{4}$ be independent, identically
distributed random variables from a population with mean $\mu$ and variance
$\sigma^{2}$.  Let $\bar{Y}=\frac{1}{4}(Y_{1}+Y_{2}+Y_{3}+Y_{4})$ denote
the average of these four random variables. \\
i) What are the expected value and variance of $\bar{Y}$ in terms of
$\mu$ and $\sigma^{2}$? \\
ii) Now, consider a different estimator of $\mu$:
\begin{eqnarray}
W=\frac{1}{8}Y_{1}+\frac{1}{8}Y_{2}+\frac{1}{4}Y_{3}+\frac{1}{2}Y_{4} \, .
\end{eqnarray}
This is a weighted average.  Show that W is also an unbiased estimator of
$\mu$.  Find the variance of W. \\
iii) Based on your answers to i) and ii) which estimator of $\mu$ do you prefer,
$\bar{Y}$ or W? \\
iv) Now consider a more general estimator of $\mu$, defined by
\begin{eqnarray}
W_{a}=a_{1}Y_{1}+a_{2}Y_{2}+a_{3}Y_{3}+a_{4}Y_{4} \, ,
\end{eqnarray}
where the $a_{i}$ are constants.  What condition is needed on the
$a_{i}$ for $W_{a}$ to be an unbiased estimator of $\mu$. \\
v) Compute the variance of the estimator $W_{a}$ from iv). \\ [8pt]

\noindent
13. You are hired by the governor to study whether a tax on liquor has decreased
average liquor consumption in your state.  You are able to obtain, for a sample
of individuals selected at random, the difference in liquor consumption (in
ounces) for the years before and after the tax.  For person $i$ who is sampled
randomly from the population, $Y_{i}$ denotes the change in liquor consumption.
Treat these as a random variable from a Normal($\mu$,$\sigma^{2}$) distribution. \\
i) The null hypothesis is that there is no change in average liquor consumption.
State this formally in terms of $\mu$. \\
ii) The alternative is that there was a decline in liquor consumption; state the
alternative in terms of $\mu$. \\
iii) Now, suppose that your sample size is n=900 and you obtain the estimates
$\bar{y}=-32.8$ and $s=466.4$.  Calculate the $t$ statistic for testing
$H_{0}$ against $H_{1}$.  Do you reject $H_{O}$ at the 5\% level?  at the
1\% level?  (Note: sample size is large so you can use the standard normal
distribution.) \\
iv) Would you say that the estimated fall in consumption is large in magnitude?
comment on the practical vs statistical significance of this estimate. \\
v) What has been implicitly assumed in your analysis about other determinants
of liquor consumption over the 2 year period in order to infer causality from 
the tax change to liquor consumption? \\ [8pt]

\noindent
14. The new management at a bakery claims that workers are now more productive
than they were under old management, which is why wages have ``generally
increased.''  Let $W_{i}^{b}$ be Worker $i$'s wage under the old management
and let $W_{i}^{a}$ be Worker $i$'s wage after the change.  The difference is
$D_{i}=W_{i}^{a}-W_{i}^{b}$.  Assume that the $D_{i}$ are a random sample from
a Normal($\mu$,$\sigma^{2}$) distribution. \\
i) Using the following data on 15 workers, construct an exact 95\% confidence
interval for $\mu$. \\
ii) Formally state the hypothesis that there has been no change in average
wages.  In particular, what is $E(D_{i})$ under $H_{0}$?  If you are hired
to examine the validity of the new management's claim, what is the relevant
alternative hypothesis in terms of $\mu=E(D_{i})$?  \\
iii) Test the null hypothesis from ii) against the stated alternative at the
5\% and 1\% levels.

\begin{center}
\begin{tabular}{rrr}
\multicolumn{1}{c}{Worker} &
\multicolumn{1}{c}{Wages Before} &
\multicolumn{1}{c}{Wages After} \\
1 & 8.30 & 9.25 \\
2 & 9.40 & 9.00 \\
3 & 9.00 & 9.25 \\
4 & 10.50 & 10.00 \\
5 & 11.40 & 12.00 \\
6 & 8.75 & 9.50 \\
7 & 10.00 & 10.25 \\
8 & 9.50 & 9.50 \\
9 & 10.80 & 11.50 \\
10 & 12.55 & 13.10 \\
11 & 12.00 & 11.50 \\
12 & 8.65 & 9.00 \\
13 & 7.75 & 7.75 \\
14 & 11.25 & 11.50 \\
15 & 12.65 & 13.00 \\
\end{tabular}
\end{center}

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