\magnification=\magstep1
\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Review Sheet, Final Exam}
\centerline{April 30, 2004}
\bigskip
\noindent
{\it General Information}
\bigskip
The final examination for this course will be given at 
2:30 a.m. on Wednesday, May 12 in our regular class room,
Swords 302.
The exam will be roughly twice the length of one of the 
midterms, but you will have the full three hour period from
2:30 pm to 5:30 pm to work on it if you need that much time.
Like the midterms, this will be an open book, open
notes exam.  Even more so than on that exam, of course,
it will be necessary to prepare carefully; you 
do not want to be flipping through the book for
every formula you need or similar examples to help you
get started on every problem.  There will not be enough
time to complete the exam if you are doing that.
\bigskip
\noindent
{\it Topics to be Covered}
\bigskip
\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)} the Central Limit Theorem, and the moment-generating
function technique we used to prove it
\item{3)} The normal approximation to binomial distributions
\item{4)} Point estimators for distribution parameters, bias, mean square error,
standard error.  Be sure you understand Table 8.1 on page 371 of the text,
where all the entries come from, and how they are used.  Also be able
to analyze estimators to determine whether they are biased or not, construct
unbiased estimators, etc.  (see Problem Set 3).  
\item{5)}  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{6)} 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 not appear on the exam.)
\item{7)} 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{8)} The connection between confidence intervals and
rejection/``acceptance'' regions for tests.
\item{9)}  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{10)}  Small sample ($t$-) tests for means and related confidence 
intervals.
\item{11)}  $\chi^2$-tests for variances and related confidence intervals.
\item{12)}  $F$-tests for ratios of variances and related confidence intervals.
\item{13)} Simple linear regression for linear models of the 
form $Y = \beta_0 + \beta_1x + \epsilon$.  Be prepared to compute
the least squares estimators for the coefficients $\beta_i$ in 
a ``small'' example (say $n \le 10$ or so) with a calculator (Maple
will not be available!).
Also know how to carry out hypothesis tests on the $\beta_i$
of the types we considered in class (see sections 11.5 and 6).  
(The material on multiple
regression, ANOVA for testing linearity of regression, etc. that
we saw in Lab Project 4 {\it will not be covered} on the final.)
\bigskip
\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.
\bigskip
\noindent
{\it Suggested Review Problems}
\bigskip
Look at the problems from the two previous
review sheets for the topics 1 - 12 above.  For the last
one (regression and hyptheses concerning the regression 
coefficients):
From Chapter 11/4, 14, 29, 51 (``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 $\beta_1 = -\alpha_1$), 73ab, 74.
\bigskip\noindent
{\it Review Session}
\bigskip
I will be happy to run a review session for the final
exam during study week.  We can discuss a time in class.
\bye

