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\centerline{Mathematics 376 -- Probability and Statistics 2}
\centerline{Midterm Examination 2}
\centerline{April 21, 2006}
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\noindent
{\it Directions}
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 Do all work in the blue exam booklet, and {\it include
all work necessary to justify your answers}. There are 100 regular
points and 10 Extra Credit points.
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\noindent
I. (20) Let $Y_1, \ldots, Y_n$ be independent samples from the distribution
with pdf containing the unknown parameter $\theta$:
$$f(y|\theta) = \cases{{1\over\theta}y^{1/\theta-1} & if $0 < y < 1$\cr
                                        0 & otherwise}$$
Determine the maximum likelihood estimator for $\theta$ using the 
$Y_i$.
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\noindent
II.  A machine shop manufactures toggle levers.  Let $d$ be 
the proportion of defectives in the shop's output.
A random sample of size $n = 150$ toggle levers
produced $6$ defectives.
\item{A)}  (10) Find a $95\%$ confidence interval for $d$ using this 
information.
\item{B)}  (10) Test the hypothesis $H_0 : d = .1$ versus
$H_a : d \ne .1$ using this data.  Take $\alpha = .02$ (probability
of Type I error).  Also give the $p$-value of your test.  
\item{C)}  (10)  For the test in part B, what is $\beta$
(probability of Type II error) if the true value of $d$ is $.03$?
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\noindent
III.  Consider the following measurements of the heat-producing
capacity of the oil produced by two fields (in billions of 
calories per ton):
$$\matrix{
{\rm Field\ 1:}& 8.26 & 8.13 & 8.35 & 8.07 & 8.34 &\cr
{\rm Field\ 2:}& 7.95 & 7.89 & 7.90 & 8.14 & 7.92 & 7.84\cr}$$
\item{A)} (10) Estimate the {\it variance} $\sigma^2$ of the heat 
producing capacity of the oil from each field.
\item{B)} (10) Test the null hypothesis $H_0: \sigma_1^2 = \sigma_2^2$
against the alternative $H_a : \sigma_1^2 \ne \sigma_2^2$ at the 
$\alpha = .05$ level.  State your conclusion
clearly and succinctly.
\item{C)} (15) Construct a two-sided $95\%$ confidence interval 
for the difference of the population mean
heat producing capacities $\mu_1 - \mu_2$.
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\noindent
IV.  (15) Let $(x_i,y_i)$, $i = 1, \ldots, n$ be a collection of
data points.  Using the matrix formulation, derive the normal equations
for the least squares estimators for the coefficients
$\beta_0,\beta_1$ in the model $Y = \beta_0 + \beta_1 x + \varepsilon$
fitting the data.
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\noindent
{\it Extra Credit} (10)  In the situation of question III part C above,
suppose you had $n = 50 = m$ measurements from each oil field
instead of $n_1 = 5$ and  $n_2 = 6$. 
What, if anything, would change in the method you would use?  Explain.
\bye
