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\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Midterm Exam I}
\centerline{March 1, 2006}
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{\it Directions:}  Do all work in the blue exam booklet.  
There are 100 regular points and 10 Extra Credit points. 
General Note:  A ``random sample'' always consists of independent
measurements from the indicated distribution.
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\noindent
I.  Assume that the calorie content
of chocolate Power Bars is normally distributed.  
A random sample of 10 chocolate Power Bars yields
calorie content values of:
$$210,220,234,227,229,240,221,228,225,219$$
\item{A)} (10) Estimate the population mean calorie content of the
chocolate Power Bars.
\item{B)} (10) The sample variance of the calorie content
values above is $S^2 = 70.68$ cal.   
Find a $99\%$ confidence interval for the population
mean calorie content, $\mu$. 
\item{C)} (10)  Find a 95\% confidence interval for the 
calorie content variance $\sigma^2$.
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II.  A researcher found that 107 out of 200 rutabaga 
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 $95\%$ confidence interval for 
the difference in the proportions of seeds that germinate
at the two temperatures. 
\item{B)} (10)  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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\noindent
III. (15) Let $Y_1,\ldots,Y_n$ denote a random sample from a population
whose density is given by 
$$f(y) = \cases{3\beta^3 y^{-4} & if $y \ge \beta$\cr
                   0 & otherwise\cr}$$
where $\beta$ is unknown.  Consider the estimator
$\widehat{\beta} = \min(Y_1,\ldots,Y_n)$.  Is $\widehat{\beta}$
biased or unbiased?
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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) Use your answer from part A to show that 
$\Sigma_X^2$ is an unbiased estimator for $\sigma^2$.
\item{C)}  (10) 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.)
\bye
