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\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Midterm Exam Im}
\centerline{March 1, 2004}
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{\it Directions:}  This is an open-book, open-notes
exam.  Do all work in the blue exam booklet.  There are 100 regular
points and 10 Extra Credit points (see back of this sheet).  Be sure to read each question
carefully before starting to work.
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General Note:  A ``random sample'' always consists of independent
measurements from the indicated distribution.
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I. (20) The amount of time (in minutes) that a dial-up
computer help desk specialist spends with one caller
is a random variable with 
a gamma distribution for $\alpha = 4$ and $\beta = 1.8$.
Estimate the probability that the mean call
time for a random sample of $64$ customers will be between
6 and 10 minutes.  Give the name of the theoretical result you are 
using to derive your estimate.
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II.  In a study conducted by the Agronomy Department at 
Cornell University, a researcher found that 111 out of 200 rutabaga
seeds germinated when they were planted in soil maintained 
at $5 {}^\circ C$, while 119 out of 200 seeds germinated when they 
were planted in soil maintained at $15 {}^\circ C$.
\item{A)} (20)  Construct a $95\%$ confidence interval for 
the difference in the proportions of seeds that germinate
at the two temperatures. 
\item{B)} (5)  Can we rely on the results of this study to 
conclude that a greater proportion of rutabaga seeds will
germinate at the higher temperature in general?  Why or why not?
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III.  A random sample of 10 healthy adult males' total serum 
cholesterol levels yielded values of
$$210,220,175,227,240,192,221,201,186,219$$
\item{A)} (5) Estimate the mean cholesterol level of this 
population.
\item{B)} (20)  Assuming the cholesterol levels are
normally distributed, find a $90\%$ confidence interval for the 
mean cholesterol level.  Explain how you decided which method
to use.
\item{C)} (5) Can we assume that the true mean cholesterol
level of this population is contained in the interval you found
in B?  Why or why not?
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IV.  Let $X_1,\ldots,X_n$ and $Y_1, \ldots, Y_n$ be two 
independent random samples of equal sizes from a normal distribution with
{\it known} mean $\mu$, but {\it unknown}
variance $\sigma^2$.  Let $\Sigma_X^2$ and $\Sigma_Y^2$
be the statistics:
$$
\Sigma_X^2 = {1\over n}\sum_{i=1}^n (X_i - \mu)^2 \qquad
\Sigma_Y^2 = {1\over n}\sum_{i=1}^n (Y_i - \mu)^2$$
\item{A)} (10) What is the distribution of $n\Sigma^2_X/\sigma^2$?
Explain.
\item{B)}  (5) What is the distribution of 
$W = {X_1-\mu \over \Sigma_Y} = {(X_1-\mu)/\sigma \over \sqrt{\Sigma_Y^2/\sigma^2}}$?
Explain. 
\item{C)} (10) Is $\Sigma_X^2$ biased or unbiased as an estimator
for $\sigma^2$?
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{\it Extra Credit} (10)  The {\it median} of a sample is 
the ``middle'' value, that is the value with the property that
half the sample values are greater than or equal to the median
and half are less than or equal to the median.  Give formulas
for computing the sample median in terms of the {\it order statistics}
of the sample.  (Note: the cases $n$ even and $n$ odd are different.)
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