\magnification=\magstep1
\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Final Examination}
\centerline{May 8, 2010}
\bigskip
\noindent
{\it Directions}
\bigskip
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 20 Extra Credit points.
\bigskip
\noindent
General Note:  A ``random sample'' always consists of independent
measurements from an indicated distribution.
\bigskip
\noindent
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 standard deviation
$\sigma = 10$.
\medskip
\item{A)} (10) What is the distribution of 
$\overline{Y} = {1\over 5}(Y_1+Y_2+Y_3+Y_4+Y_5)?$
Why? 
\item{B)} (10) What is the distribution of 
$U = {(Y_1 - 4)^2 + (Y_2 - 4)^2 + (Y_3 - 4)^2 \over 100}?$
Why?
\item{C)} (10)  What is the distribution of 
$V = {\sqrt{3}(Y_4 - 4)\over 10 \sqrt{U}},$
where $U$ is as in part B?  Why?
\item{D)}  (10)  How would you determine the PDF for the sample maximum $Y_{(5)}$?  
(Note:  The CDF for a normal random variable is not an elementary function; just
give a ``recipe'' for how it might be computed.)
\bigskip
\noindent
II.  A random variable $Y$ is said to have a {\it log-normal}
distribution with parameters $\mu,\sigma$ if its pdf has the form
$$f(y) = \cases{
{1\over y\sqrt{2\pi\sigma^2}} e^{-(\ln(y) - \mu)^2/(2\sigma^2)} & if $y > 0$\cr
 &\cr
0 & otherwise\cr}$$
\medskip
\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 known 
$\sigma = 1$.  Find the maximum-likelihood estimator for $\mu$.  
\item{C)} (5)  Is your estimator from part B biased or unbiased?
Why?
\bigskip
\noindent
III.  (15) Solid copper produced by melting powdered ore is tested
for ``porosity'' (the volume fraction due to air bubbles).  
A sample of $n_1 = 40$ porosity measurements made in one lab
has $\overline{y}_1 = .25$ and $s_1^2 = .001$.  A second set of $n_2 = 50$
measurements is made using identical ore in a second lab, yielding 
$\overline{y}_2 = .17$ and $s_2^2 = .002$.  
Find a $95\%$ confidence interval for $\mu_1 - \mu_2$, 
the difference of the population mean porosity measurements from 
the two labs.  What conclusion can you draw from your interval?
\bigskip
\noindent
IV.  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.  Out of a random sample
of 50 ingots of a certain grade of steel from one manufacturer, 
the Rockwell hardness index was greater than 62 in 31 of the ingots.  
\medskip
\item{A)} (15) The manufacturer claims that at least 75\% of the 
ingots of this type of it produces steel will have Rockwell hardness 
index greater than 62.  Is there sufficient evidence to refute this claim?  Use a test at the $\alpha = .01$ level.
\item{B)} (15) Using the rejection region you found in part A,
compute the Type II error probability $\beta$ of your test 
if it is actually true that $65\%$ of the ingots have hardness 
index greater than 62.  
\bigskip
\noindent
V.  Consider the following measurements of the weights 
of yields of two breeds of apple trees (in kilograms):
$$\matrix{
{\rm Breed\ 1:}& 80.6 & 80.3 & 81.5 & 80.7 & 80.4 &\cr
{\rm Breed\ 2:}& 79.5 & 79.9 & 81.0 & 79.4 & 79.2 & 81.4\cr}$$
Assume the measurements come from normal populations.  
\bigskip
\item{A)} (20) Estimate the {\it variances} $\sigma_1^2$ and $\sigma_2^2$ of the yields of the two breeds.
Is there evidence to suspect that $\sigma_2^2 > \sigma_1^2$? Explain.  
\item{B)} (15) Test the null hypothesis $H_0: \mu_1 = \mu_2$
against the alternative $H_a : \mu_1 \ne \mu_2$.
Estimate the $p$-value of your test and state your conclusion
clearly and succinctly.
\bigskip
\noindent
VI.  The following table gives measurements of the amount
of sodium chloride that dissolved in 100 grams of water
at various temperatures in a chemistry experiment. 
$$\matrix{
x {\rm\ (degrees\ C)} & y {\rm\ (grams)}\cr
      0               &       7.3\cr
      15              &       13.0\cr
      30              &       23.3\cr
      45              &       30.7\cr
      60              &       39.7\cr
      75              &       47.7\cr}$$
\item{A)}  (20)  Find the equation of the least squares 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.  
\bigskip
\noindent
VII. (20) (A ``Thought Question'') Does a ``statistically significant'' result where we reject some $H_0$ 
mean that $H_0$ is {\it far from being true}?  Answer intuitively first.  Then answer
the following:  Suppose each individual we draw from a population
has either property $A$ or property $B$ (but not both). 
Let $p$ be the proportion of the population that has property $A$.
We test $H_0: p = 1/2$ versus $H_a : p > 1/2$ with large-sample tests 
with $n = 100, 1000, 10000, 100000$.  What must the observed proportion of sampled 
individuals with property $A$ be in order to reject $H_0$ at the $\alpha = .05$ 
level in each case?  How does this square with what you said at first?
\bigskip
\noindent
{\it Extra Credit} (20)  Recall that in the ``Big Theorem'' in the multiple
regression case, we said that in the entries $c_{ij} \sigma^2$ of the covariance matrix
of the least squares estimators, the $c_{ij}$ were the entries of the matrix $(X^tX)^{-1}$.
Show this is true by direct computation for the $X$ from a simple linear model of the form 
$Y = \beta_0 + \beta_1 x + \epsilon$.   
\bigskip
\centerline{\it Have a safe, enjoyable, and productive summer!}
\bye
