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\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Midterm Exam I}
\centerline{February 27, 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.  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.  A random sample of 10 chocolate Power Bars yielded
calorie content values of
$$210,220,234,227,229,240,221,228,225,219$$
\item{A)} (5) Estimate the mean calorie content of this type
of chocolate Power Bar.
\item{B)} (15) The sample variance of the calorie content
values above is $S^2 = 70.68$ cal.  Assuming the calorie content
of the whole population of chocolate Power Bars is 
normally distributed, find a $99\%$ confidence interval for the 
mean calorie content.  Explain how you decided which method
to use.
\item{C)} (5) Can we assume that the true mean calorie content of this type
of chocolate Power Bar is contained in the interval you found
in B?  Why or why not?
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II.  In a study conducted by the Horticulture Department at 
Virginia Tech, a researcher found that 107 out of 200 broccoli 
seeds germinated when they were planted in soil maintained 
at $5 {}^\circ C$, while 123 out of 200 seeds germinated when they 
were planted in soil maintained at $15 {}^\circ C$.
\item{A)} (20)  Construct a $90\%$ 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 broccoli seeds will
germinate at the higher temperature in general?  Why or why not?
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III.  
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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)} (10) Use your answer from part A to show that 
$\Sigma_X^2$ is an unbiased estimator for $\sigma^2$.
\item{C)}  (5) What is the distribution of $\Sigma_X^2/\Sigma_Y^2$?
Explain. 
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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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