\magnification=\magstep1
\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Review Sheet -- Exam II}
\centerline{April 2, 2004}
\bigskip
\noindent
\bigskip
\noindent
{\it General Information}
\bigskip
As announced in the course syllabus, the second midterm exam this semester
will be given in class on {\it Friday, April 16}.  The exam will
cover the material we have discussed since the last exam, 
starting from the material on confidence intervals for variances
from section 8.9 (see Lab Project 2), and going through the 
material we covered in Chapter 10 on hypothesis testing, culminating
in Lab Project 3.  There is 
a more detailed breakdown of topics given below.
\bigskip
\noindent
{\it Format}
\bigskip
This will be an {\it open book, open class notes, open problem sets/solutions exam}.  
But you should still prepare carefully for the exam and 
understand the key concepts we have talked about.  Know
how to apply the different techniques we have studied and how
to select the most appropriate method when there are several
possible choices.
If you are flipping through the book to look for {\it every} formula 
you need or if you need to find examples similar to the test questions
to get started on a problem, you {\it will not be able to finish the exam 
in 50 minutes}.
\bigskip
\noindent
{\it Topics}
\bigskip
\noindent
The topics to be covered (not in chronological order,
but according to logical connections):
\bigskip
\item{1)} The method of moments and the method of maximum
likelihood for deriving estimators.  (Some of the other material
we discussed in Chapter 9 on consistency of estimators, 
sufficient statistics, etc. is mainly background theoretical information
for your general mathematical education and will not feature in the exam
questions.)
\item{2)} 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{3)} The connection between confidence intervals and
rejection/``acceptance'' regions for tests.
\item{4)}  Large sample ($Z$-) tests and related confidence intervals
for means and proportions (Note: some of this overlaps material
from Midterm 1!).  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{5)}  Small sample ($t$-) tests for means and related confidence intervals.
\item{6)}  $\chi^2$-tests for variances and related confidence intervals.
\item{7)}  $F$-tests for ratios of variances and related confidence intervals.
\bigskip
\noindent
{\it Comment:}  As you should be able to tell, the justifications for 
the methods we have developed here depend heavily on the 
probability topics we learned last semester, as well as the sampling
distribution theory ($Z, \chi^2, t, F$ distributions, etc.) and
the Central Limit Theorem from Exam 1.  
If you are feeling ``rusty'' on any of that, start by reviewing that 
material.
\bigskip
\noindent
{\it Review Session}
\bigskip
I will be happy to run a pre-exam review session if there is interest.
The evening of Wednesday, April 14 would be best, but
other times might be possible too. 
\bigskip
\noindent
{\it Suggested Review Problems}
\bigskip
From the text: Chapter 9/62, 63, 64, 73 (ignore the Hint -- just work it out!),
75, 76 a,b, Chapter 10/3 (this continues problem 2, which we did as an example in 
class), 103, 105, 106, 109, 111, 112
\bigskip
\noindent
{\it Sample Exam}
\bigskip
\noindent
Disclaimer: A reasonable 1 hour exam cannot cover every topic
we have discussed in this section of the course (in particular
every type of estimation and hypothesis testing problem we
have seen).  You should be prepared for all the possibilities though.
The following problems indicate the approximate
level of difficulty and cover a possible subset of the 
topics on the upcoming exam; the actual exam
problems may differ substantially from these and might deal with
different situations.
\bigskip
\noindent
Comments:  Any general formula we have studied can be
used without comment (i.e. you don't need to rederive it in 
you solution).
\bigskip
\noindent
I.  Let $Y_1, \ldots, Y_n$ be independent samples from the distribution
with pdf containing the unknown parameter $\theta$:
$$f(y|\theta) = \cases{{1\over\theta}y^{(1-\theta)/\theta} & if $0 < y < 1$\cr
                                        0 & otherwise}$$
\item{A)}  What is the likelihood function 
$L(y_1,\ldots,y_n|\theta)$?
\item{B)}  Determine the maximum likelihood estimator for $\theta$.
\bigskip
\noindent
II.  Let $X$ and $Y$ be the forces required to pull two different
studs out of an automobile window.  Assume that the distributions
are both normal with $X\sim  N(\mu_X,\sigma^2_X)$ and 
$Y\sim  N(\mu_Y,\sigma^2_Y)$.  
\item{A)} Assuming that $\sigma_X^2 = \sigma_Y^2$, and you 
have $n = 10$ $X_i$ measurements and $m = 10$ $Y_j$ measurements,
specify a test statistic and rejection region for testing
$H_0: \mu_X = \mu_Y$ against $H_a: \mu_X \ne \mu_Y$, if 
the test is to have level $\alpha = .05$.  
\item{B)}  The following $n = 10$ measurements
of $X$:
$$111, 120, 139, 136, 138, 149, 143, 145, 111, 123$$
and $m = 10$ measurements of $Y$:
$$152, 155, 133, 134, 119, 155, 142, 146, 157, 149$$
were obtained.  Calculate your test statistic from part
A) and state your conclusion.
\item{C)}  Compute a two-sided $95\%$ confidence interval 
for $\mu_X - \mu_Y$ and state the relation between your answer
here and your answer for part B.
\item{D)}  What is the approximate $p$-value of your test in part B?
\item{E)}  Test whether the assumption $\sigma_X^2 = \sigma_Y^2$
seems valid, using an appropriate test with level $.05$.
\item{F)}  Suppose you had $n = 50 = m$ measurements of $X$ and $Y$ instead 
of $n = 10 = m$.  What, if anything, would change in your answer to part A?
Explain.  
\item{G)}  Assuming $m=n$ can be chosen arbitrarily, how large
would be need to take them to obtain a test as in part F with 
$\beta = .05$, assuming $\mu_X - \mu_Y = 10$?
\bigskip
\noindent
III.  (Essay)  In intuitive terms we might say ``if the 
test statistic turns out to be outside the rejection region,
then we accept $H_0$.''  Is that precisely correct, though?
Should we take the results of the test as indicating $H_0$
is definitely true in that case?
\bye
