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\centerline{Mathematics 400 -- Directed Readings in Probability and Statistics 2}
\centerline{Midterm Examination 2}
\centerline{April 22, 2010}
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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. Let $Y_1, \ldots, Y_n$ be independent samples from the distribution
with pdf containing the unknown parameter $\theta > 0$:
$$f(y|\theta) = \cases{{1\over\theta}y^{1/\theta-1} & if $0 < y < 1$\cr
                                        0 & otherwise}$$
\item{A)} (10)  Determine the method of moments estimator for $\theta$ using
the $Y_i$.  
\item{B)} (15)  Determine the maximum likelihood estimator for $\theta$ using the 
$Y_i$.
\item{C)} (5)  Suppose you have $n = 5$ data values $y_1,\ldots,y_5 = .78, .80, .70, .84, .47$.
What are the two estimates of $\theta$ given by your answers from parts A and B?
\item{D)}  (10)  Suppose you set up a test of the hypothesis $H_0:  \theta = 1$ versus
the alternative $H_a:  \theta > 1$ using $Y_1, Y_2$ (exactly two of the samples).  
If you reject $H_0$ when $Y_1 + Y_2 > .8$, what is the Type I error probability, $\alpha$?
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\noindent
II.  A shop manufactures O-rings for the Space Shuttle booster rockets
for NASA.  Let $d$ be the proportion of defectives in the shop's output.
A random sample of size $n = 70$ O-rings produced $6$ defectives.
\item{A)}  (15) 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{B)}  (15)  For the test in part A, 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 natural gas produced by two fields of gas wells
(in calories per cubic meter):
$$\matrix{
{\rm Field\ 1:}& 8.26 & 8.13 & 8.35 & 8.07 & 8.34 &\cr
{\rm Field\ 2:}& 7.95 & 7.89 & 7.90 & 8.14 & 7.92 & 7.84\cr}$$
Let $\mu_i$ ($i = 1,2$) be the population mean heat-producing capacity
of the natural gas from field $i$.  
\item{A)} (15)  What assumptions do you need to make in order 
to use the appropriate test of $H_0: \mu_1 = \mu_2$ versus
$H_a: \mu_1 \ne \mu_2$?  How could you determine whether 
it is reasonable to assume those assumptions are satisfied?  (You do not need to 
carry out a test here -- just say how the test would be set up.)
\item{B)}  (10)  Carry out the test from part A with $\alpha = .05$ 
and state your conclusion clearly and succinctly.
\item{C)} (5) Construct a two-sided $95\%$ confidence interval 
for the difference of the population mean heat producing capacities $\mu_1 - \mu_2$.
How is this related to your answer in B?  
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\noindent
{\it Extra Credit} (10)  In the situation of question I, is there one
of the two estimators that you derived that is clearly ``better''
to use?  Explain how you could answer a question like this.
\bye
