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\centerline{Mathematics 376 -- Mathematical Statistics}
\centerline{Midterm Exam 2}
\centerline{April 26, 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.  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
$$f(y|\theta) = \cases{{4y^3\over\theta^4} & if $0\le y\le \theta$\cr
                            0 & otherwise.\cr}$$
\item{A)} (10) Determine the method of moments estimator for
$\theta$.
\item{B)} (10) Determine the maximum likelihood estimator for 
$\theta$.
\item{C)} (5) Is $U = \sum_i \ln(Y_i)$ a sufficient statistic for $\theta$?
Why or why not?  
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II.  Assume that the hardness of iron bars produced by one
plant is normally distributed with mean $\mu$
and variance $\sigma^2$.  A random sample of $n = 7$ bars 
from the output of the plant 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 $H_0 : \mu = 172$ versus the alternative 
$H_a : \mu \ne 172$ using an appropriate test with $\alpha = 0.01$.
\item{B)} (10)  How is your answer from A related to the $99\%$ 
confidence interval for $\mu$ that we learned to compute in the first
section of the course?
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III.  In this problem, let $S$ represent the proportion of women
who agree that men are basically selfish and egotistical.  
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\item{A)} (15)  Suppose that surveys done 20 years ago showed that $S$ was 
$0.40$ at that time, 
but a new survey finds that $Y = 1260$ out of a random sample of $n = 3000$ 
women said that men are basically selfish and egotistical.
Is there sufficient evidence to say the proportion $S$ has 
increased?  Give the null and alternative
hypotheses for a suitable test, carry out the analysis, estimate the 
attained significance level ($p$-value) and state your
conclusion succinctly and clearly.  
\item{B)} (15) Suppose that the current value of $S$ is actually
$S = .43$ and the test in part A was carried out to yield $\alpha = 0.05$.
How large would $n$ have to be in a sample like the 
one from part A to ensure that $\beta$ (the probability of 
a Type II error) is $.1$ or less?  
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IV.  (15) Let $(x_i,y_i,z_i)$, $i = 1, \ldots, n$ be a collection of
data points.  Using the matrix formulation, derive the normal equations
for the least squares estimators for the coefficients
$\beta_0,\beta_1,\beta_2$ in the model $Z = \beta_0 + \beta_1 x + \beta_2 y + \varepsilon$
fitting the data.  Express in the form $A\beta = b$ where $A$ is 
a matrix and $b$ is a vector (not matrix products). 
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{\it Extra Credit ``Thought Questions'' -- Answer in 2 or 3 complete sentences:}
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\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 exactly?  Explain using the case of a test on a single mean.
\item{B)} (5)  For the basic small-sample test for equality of means,
we need to assume the two samples came from normal populations
with equal variances.  How would you determine whether applying
that test was appropriate?

\bye
