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\centerline{Mathematics 376 -- Mathematical Statistics}
\centerline{Makeup Midterm Exam I -- March 19, 2012}
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{\it Directions:}  Do all work in the blue exam booklet.  
There are 100 regular points and 10 Extra Credit points (question on back
of this sheet). 
General Note:  A ``random sample'' always consists of independent
measurements from the indicated distribution.
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\noindent
I.  Let $X_1,\ldots,X_n$ and $Y_1, \ldots, Y_n$ be two 
independent random samples of equal sizes from a population
with a normal distribution having a
{\it known} mean $\mu$, but {\it unknown}
variance $\sigma^2$.  Let $\Sigma_X^2$ and $S_Y^2$
be the statistics:
$$
\Sigma_X^2 = {1\over n}\sum_{i=1}^n (X_i - \mu)^2 \qquad
S_Y^2 = {1\over n-1}\sum_{i=1}^n (Y_i - \overline{Y})^2.$$
Answer the following questions, giving complete explanations:
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\item{A)} (5)  What is the distribution of ${X_1 - \mu\over \sigma}$?
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\item{B)} (5) What is the distribution of 
$W = {n\Sigma^2_X\over \sigma^2}?$
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\item{C)}  (10) What is the distribution of 
$V = {\Sigma_X^2\over S_Y^2}?$
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\item{D)}  (10) What is the distribution of 
$U = { Y_1 - \mu\over \Sigma_X}?$
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\noindent
II.  Assume that the calorie content
of Nature Valley granola bars is normally distributed.  
A random sample of 10 granola bars yields
calorie content values of:
$$210,220,234,227,229,240,221,228,225,219$$
\item{A)} (20) Find a $99\%$ confidence interval for the population
mean calorie content, $\mu$, for this type of granola bar.  
Explain how you know which method to use.  
\item{B)} (10)  Find a $90\%$ confidence interval for the population
calorie content standard deviation $\sigma$.
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\noindent
III.  A researcher found that in two independent random samples,
107 out of 200 apple seeds germinated when they were planted in soil maintained 
at $5 {}^\circ C$, while 130 out of 200 seeds germinated when they 
were planted in soil maintained at $15 {}^\circ C$.
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\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)  Do the results of this study suggest that 
a greater proportion of apple seeds will germinate at the 
higher temperature in general?  Why or why not?
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\noindent
IV. (15) Let $Y_1,\ldots,Y_n$ denote a random sample from a 
uniformly distributed population on an interval $[\theta,2\theta]$
where $\theta$ is unknown.  Consider the estimator
$\widehat{\theta} = {1\over 2}\cdot \max(Y_1,\ldots,Y_n)$.  Is $\widehat{\theta}$
biased or unbiased?
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\noindent
{\it Extra Credit} (10)  In honor of $\pi$-day, here is the fact
behind one of the demos that we were running in the McBrien 
Student Common Room yesterday.  Suppose we drop a rod 1 unit long 
so that its midpoint lands at a point with $y$-coordinate
uniformly distributed on the interval $[0,1/2]$, and the 
angle between the horizontal and the rod is also uniformly
distributed on the interval $[0,\pi]$.  Show that the
probability that the rod intersects the $x$-axis is ${2\over \pi}$.
\bye
