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\centerline{Mathematics 376 -- Probability and Statistics 2}
\centerline{Review Sheet, Final Exam}
\centerline{April 28, 2006}
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\noindent
{\it General Information}
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The final examination for this course will be given at 
8:30 a.m. on Saturday, May 6 in our regular class room,
Swords 328.
The exam will be roughly twice the length of one of the 
midterms, but you will have the full three hour period from
8:30 am to 11:30 am to work on it if you need that much 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 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.  
\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 not appear on the 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.
\item{10)}  $F$-tests for ratios of variances and related confidence intervals.
\item{11)}  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 
(Maple will not be available!).
Also know how to carry out hypothesis tests on the $\beta_i$ in 
this case as in some of the questions from Problem Set 8.
\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}
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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}
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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.
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{\it Important Note about Scheduling and a Review Session}
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I will be attending a conference in Linz, Austria
Monday through Wednesday next week (May 1,2,3).  
Since Lab 5 completes the material for the course, and there
is no other unfinished business, it will
make sense just to cancel the last scheduled class meeting (May 1).
I will be happy to run a review session for the final
exam after I return.  Either morning or afternoon on 
Thursday May 4 are possible.   We can discuss a time in class.
\bye

