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\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Makeup Midterm Exam II}
\centerline{April 19, 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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I.  Let $Y_1,\ldots,Y_n$ be a random sample from a population
with density function
$$f(y|\theta) = \cases{{3\theta^3\over y^4} & if $\theta\le y < +\infty$\cr
                            0 & otherwise\cr}$$
containing the parameter $\theta > 0$.
\item{A)} (10) Determine the method of moments estimator for
$\theta$.
\item{B)} (10) Determine the maximum likelihood estimator for 
$\theta$.
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II.  In this problem, let $p$ represent the proportion of women
who agree that men are basically selfish and egotistical.  
Suppose that in the past $p$ was $0.40$, but we assert that $p$ 
has now increased.  
\item{A)} (15) State the null and alternative hypotheses, test statistic
and rejection region for a test of this assertion
with $\alpha = .05$, assuming a large sample size.  Sketch
a standard normal pdf and illustrate the rejection region.
\item{B)} (10) The Detroit {\it Free Press} reported in 1990 
that $Y = 1260$ women
out of a random sample of $n = 3000$ agreed with the 
statement that men are basically selfish and egotistical.
What conclusion do you draw from your test in part A with this data?
\item{C)} (10) Suppose that the current value of $p$ is actually
$p = .43$.  Determine $\beta$ ($=$ probability of a Type II  
error in your test) if $n = 3000$ as in part B.
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III.  Let $X$ be the Brinell hardness measurement of an iron
bar, and assume $X$ is normally distributed with mean $\mu$
and variance $\sigma^2$.  A random sample of 7 bars 
is measured to determine whether the average
hardness meets the desired figure $\mu_0 = 172$. 
\item{A)} (20)  The following measurements:
$167, 174, 179, 164, 163, 160, 168$ were obtained.
Test whether the data indicate $\mu < 172$ using $\alpha = .1$
and state your conclusion.
\item{B)} (15)   Test the null hypothesis $H_0 : \sigma^2 = 40$
against the alternative hypothesis $H_a : \sigma^2 > 40$
using an appropriate test with $\alpha = .1$, and state your conclusion.
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IV.  {\it ``Thought Questions'' -- Answer in 2 or 3 complete sentences:}
\item{A)} (5)  Exactly why is it the case that
a $Z$-test is not appropriate
when the sample size is small and population variance is not
known?  Explain using the case of a test on a single mean.
\item{B)} (5)  Suppose you do two different tests:
an upper-tail test and a two-tail test.  How would the 
$p$-values of those tests compare if you use the same data for each?  
Explain.
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{\it Extra Credit} (10) What is the standard error for the method
of moments estimator from problem I part A?  How would you use that 
information to determine which of the two estimators in problem I
is superior?
\bye