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\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Final Examination}
\centerline{May 12, 2004}
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{\it Directions}
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This is an open-book, open-notes
exam.  Do all work in the blue exam booklet, and {\it include
all work necessary to justify your answers}.  Be sure to read each question
carefully before starting to work.  There are 200 regular
points and 10 Extra Credit points.
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\noindent
General Note:  A ``random sample'' always consists of independent
measurements from the indicated distribution.
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I.  Let $Y_1,Y_2,Y_3,Y_4,Y_5$ be a random sample from a 
normal distribution with mean $\mu = 4$ and variance $\sigma^2 = 100$.
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\item{A)} (6) What is the distribution of 
$\overline{Y} = {1\over 5}(Y_1+Y_2+Y_3+Y_4+Y_5)?$
Why? 
\item{B)} (7) What is the distribution of 
$U = {(Y_1 - 4)^2 + (Y_2 - 4)^2 + (Y_3 - 4)^2 \over 100}?$
Why?
\item{C)} (7)  What is the distribution of 
$V = {\sqrt{3}(Y_4 - 4)\over 10 \sqrt{U}},$
where $U$ is as in part B?  Why?
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II.  Let $Y_1,\ldots,Y_{50}$ be a random sample of measurements
of the proportion of impurities in bauxite ore samples.  Each
$Y_i$ is a random variable with pdf 
$$f(y) = \cases{4y^3 & if $0 \le y \le 1$\cr
                  0  & elsewhere\cr}$$
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\item{A)} (15) Compute $E(Y_i)$ and $V(Y_i)$.
\item{B)} (15) Estimate $P(\overline{Y} > .85)$
(the probability that the sample mean
is at least $.85$) using the Central Limit Theorem.
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\noindent
III.  A random variable $Y$ is said to have a {\it log-normal}
distribution with parameters $\mu,\sigma^2$ if its pdf has the form
$$f(y) = \cases{
{1\over y\sqrt{2\pi\sigma^2}} e^{-(\ln(y) - \mu)^2/2\sigma^2} & if $y > 0$\cr
0 & otherwise\cr}$$
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\item{A)} (10) Compute $E(\ln(Y))$ for a log-normal random variable.
(Hint: Set up the integral, then change variables.)
\item{B)} (10) Let $Y_1,\ldots,Y_n$ be a random sample
from a log-normal distribution with unknown $\mu$ and 
$\sigma^2 = 1$.  Find the maximum-likelihood estimator
for $\mu$.  
\item{C)} (10)  Is your estimator from part B biased or unbiased?
Why?
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IV.  Solid copper produced by melting powdered ore is tested
for ``porosity'' (the volume fraction due to air bubbles).  
A sample of $n_1 = 4$ porosity measurements made in one lab
has $\overline{y}_1 = .25$ and $s_1^2 = .001$.  A second set of $n_2 = 5$
measurements is made using identical ore in a second lab, yielding 
$\overline{y}_2 = .17$ and $s_2^2 = .002$.  
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\item{A)} (15) Find a $95\%$ confidence interval for $\mu_1 - \mu_2$, 
the difference of the population mean porosity measurements from 
the two labs.
\item{B)} (15) Is there sufficient evidence at the $\alpha = .05$ 
level to conclude that $\mu_1 \ne \mu_2$?   Explain.
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V.  The Rockwell hardness index for steel is determined by 
pressing a diamond point into the metal with a specified force
and measuring the depth of penetration.  For a random sample
of 50 pieces of a certain grade of steel, the Rockwell
hardness index averaged 62, with sample standard deviation 8.  
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\item{A)} (15) The manufacturer claims that this steel has
average hardness index (at least) 64.  Is there sufficient
evidence to refute this claim?  Use a lower-tail test 
at the $\alpha = .01$ level.
\item{B)} (15) Using the rejection region you found in part A,
compute the Type II error probability $\beta$ if the actual
mean hardness index is $\mu_a = 60$.  
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VI.  Consider the following measurements of the heat-producing
capacity of the coal produced by two mines (in millions of 
calories per ton):
$$\matrix{
{\rm Mine\ 1:}& 8260 & 8130 & 8350 & 8070 & 8340 &\cr
{\rm Mine\ 2:}& 7950 & 7890 & 7900 & 8140 & 7920 & 7840\cr}$$
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\item{A)} (15) Estimate the {\it variance} of the heat 
producing capacity of the coal from each mine.
\item{B)} (15) Test the null hypothesis $H_0: \sigma_1^2 = \sigma_2^2$
against the alternative $H_a : \sigma_1^2 \ne \sigma_2^2$.
Estimate the $p$-value of your test and state your conclusion
clearly and succinctly.
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VII.  The following table gives measurements of the amount
of sodium chloride which dissolved in 100 grams of water
at various temperatures. 
$$\matrix{
x {\rm\ (degrees\ C)} & y {\rm\ (grams)}\cr
      0               &       7.3\cr
      15              &       13.0\cr
      30              &       23.3\cr
      45              &       30.7\cr
      60              &       39.7\cr
      75              &       47.7\cr}$$
\item{A)}  (15)  Find the equation of the regression line
for this data set.
\item{B)}  (15)  Is there sufficient evidence to say
that $\beta_1 > .45$?  Explain, using the $p$-value of an
appropriate test.  
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\noindent
{\it Extra Credit}  (Short Essay -- no more than
a paragraph, please!) (10) Exactly what does it mean when
we reject a null hypothesis in a statistical test?  Are we 
saying that the null hypothesis is definitely false?  
Explain using the quantities $\alpha$ and $\beta$.  Also, how
does the ``strength'' of the results of a test depend on the values of 
$\alpha$ and $\beta$?
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\centerline{\it Have a safe, enjoyable, and productive summer!}
\bye