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\centerline{Mathematics 376 -- Mathematical Statistics}
\centerline{Review Sheet, Final Exam}
\centerline{May 7, 2012}
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\noindent
{\it General Information}
\bigskip
The final examination for this course will be given at 11:30 a.m. 
on Wednesday, May 16 in our regular class room, Swords 302.
The exam will be roughly one and a half times the length of 
one of the midterms.  You will have the full two and one half hour period from
11:30 am to 2:00 pm to work on it if you need that much time.
{\it Important Note:  Because others may need the room and because some of
you may have exams after this one, I will not be able
to give you extra time this time.}
As was true for the midterms, I will let you bring an 8.5 by 11
sheet of paper to the exam containing any information you 
want and you may consult it at any time.  Copies of any 
necessary tables will also be provided.    Even more so than on the midterms, 
of course, it will be necessary to prepare carefully; you don't
want to have to search for
every formula you need or a similar examples to help you
get started on every problem.  You will be pressed
for time to complete the exam if you are doing that.
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\noindent
{\it Topics to be Covered}
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\item{1)}  Sampling distributions related to the normal distribution
($\chi^2, t, F$ distributions) -- know how to 
tell when a random variable has one of these distributions, how 
to use the tables for each in the text, etc.
\item{2)} Point estimators for distribution parameters, bias, mean square error,
standard error.  Be sure you understand Table 8.1 on page 397 of the text,
where all the entries come from, and how they are used.  {\it Be aware that
this table covers only the large sample case.
The confidence intervals and tests for means and difference of means  
in the small sample case are slightly different because of the hypotheses
needed to assure the relevant statistic has a $t$-distribution. Review 
Section 8.8 if this is not clear.}  Also be able
to analyze estimators to determine whether they are biased or not, construct
unbiased estimators, etc.  
\item{3)}  The pdf's for order statistics (especially the 
sample maximum and minimum), and how they can be used for estimation
problems, especially in conjunction with:
\item{4)} The method of moments and the method of maximum
likelihood for deriving estimators.  (The other material
we discussed in Chapter 9 on consistency of estimators, 
sufficient statistics, etc. will {\it not} appear on this exam.)
\item{5)} Hypothesis testing -- the general concepts:
null hypothesis, alternative hypothesis, test statistic,
rejection region, Type I error probability ($= \alpha$, or level of 
test), Type II error probability ($= \beta$), attained significance level
($p$-value of a test), interpretation of results.
\item{6)} The connection between confidence intervals and
rejection/``acceptance'' regions for tests.
\item{7)}  Large sample ($z$-) tests and related confidence intervals
for means and proportions.  Questions here might also 
ask you to design tests with a given $\alpha$-value to achieve a 
certain $\beta$-value by selecting sample size appropriately.
\item{8)}  Small sample ($t$-) tests for means and related confidence 
intervals.
\item{9)}  $\chi^2$-tests for variances and related confidence intervals;
$F$-tests for ratios of variances and related confidence intervals.
\item{10)}  Least squares estimation for linear models of the 
form $Y = \beta_0 + \beta_1x_1 + \cdots + \beta_k x_k + \varepsilon$.  
The kinds of questions I might ask here will be of two types:  
\itemitem{a)}  
Be prepared to compute the least squares estimators for the coefficients 
$\beta_i$ in a ``small'' simple regression ($k = 1$)
example (say with $n \le 10$ or so) with a calculator 
({\tt R} will not be available!).
Also know how to carry out hypothesis tests on the $\beta_i$ in 
this case.
\itemitem{b)}  I might also ask you to set up the normal equations for
the least squares estimators in a multiple regression ($k \ge 2$)
example or answer some general questions about hypothesis testing
in that situation.  I will not ask any computational questions
there, however.
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\noindent
{\it Suggestions on How to Study}
\bigskip
Start by reading the above list of topics carefully.
If there are terms there that are unfamiliar or for which
you cannot give the precise definition review them.
Reread the class notes.  {\it Everything on 
the final will be similar to something we have discussed at
some point this semester.}  Also look back over your graded
problem sets and exams.  If there are problems that you 
did not get the first time around, try them again now.  Then
go through the suggested problems from the 
review sheets.  If you have worked these out previously,
it is not necessary to do them all again.  But try a representative
sample ``from scratch''.  Don't just look over your old
solutions and nod your head if it looks familiar. 
Practice thinking through the logic of how the solution is derived again.
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\noindent
{\it Suggested Review Problems}
\bigskip
Look at the problems from the two previous
review sheets for the topics 1 - 9 above.  For the last
one (regression and hyptheses concerning the regression 
coefficients):
From Chapter 11/31, 63 (``linearize the model''
means to take the logarithm of both sides
of $E(Y) = \alpha_0e^{-\alpha_1 x}$ as 
$\ln(E(Y)) = \ln(\alpha_0) - \alpha_1 x$ and do the regression
with the data points $(x_i,\ln(y_i))$ to estimate $\beta_0 = \ln(\alpha_0)$
and $\beta_1 = -\alpha_1$), 69, 72 (Exam question could be ``How would you
set up a test of $H_0: \beta_2 = 0$ vs.  $H_a: \beta_2 \ne 0$?'')
\bigskip\noindent
{\it Review Session}
\bigskip
If there is interest, I would be happy to run a review
session after the end of classes.  We can discuss this
in our last class meeting.
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\noindent
{\it Sample Exam Questions}
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\noindent
General Notes: 
\item{(1)} A ``random sample'' always consists of independent
measurements from an indicated distribution.  
\item{(2)} This is the exam from the Spring 2010 offering of 
MATH 376; this year's exam will be a bit shorter because
the exam period is now only 2.5 hours long.  
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\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.)
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\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?
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\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?
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\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.  
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\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.  
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\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.
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\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.  
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\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 
$Y = \beta_0 + \beta_1 x + \epsilon$.   
\bye

