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\centerline{Mathematics 400 -- Directed Readings in Probability and Statistics 2}
\centerline{Review Sheet -- Exam II}
\centerline{April 14, 2010}
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{\it General Information}
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As announced in the course syllabus, the second midterm exam this semester
will be given on {\it Thursday, April 21}, probably in the evening.  
The exam will cover the material we have discussed since the last exam, 
starting from the material in Chapter 9 that we covered in class, 
including the material from Chapter 10 on hypothesis testing.
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{\it Format}
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As on the last exam, you may prepare a sheet (single side
of standard 8 1/2 by 11 inch paper) of formulas and 
any other information you want to include and consult
it during the exam at any time.  
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 need to find examples similar to the test questions
to get started on a problem, the exam will take much longer
to complete than is necessary.
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{\it Topics}
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The topics to be covered (not in chronological order,
but according to logical connections):
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\item{1)} The method of moments and the method of maximum
likelihood for deriving estimators.  
\item{2)}  Consistency of estimators (know the theorem giving
a sufficient condition for consistency), sufficient statistics (know
the factorization criterion).   
\item{3)} 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{4)} The connection between confidence intervals and
rejection/``acceptance'' regions for tests.
\item{5)}  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{6)}  Small sample ($t$-) tests for means and related confidence intervals.
\item{7)}  There may be a part of a question dealing with 
$\chi^2-$ and $F$-tests for variances and ratios of variances and related 
confidence intervals.  This is something we did not discuss directly in 
class, but it is covered in section 10.9, and the idea should be relatively
clear from what we did before with confidence intervals.   See question III B
below for an example.
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{\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.  If you are feeling ``rusty'' on any of that, 
start by reviewing that material.
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{\it Review Session}
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We will review for the exam in the regular class on Tuesday, April 19, and 
I will be available for questions during our usual class time on 
Thursday (in place of a regular class meeting).  
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{\it Suggested Review Problems}
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From the text: 
\item{$\bullet$}  Chapter 9/3,15,42,53,69,77,91,93; 
\item{$\bullet$}  Chapter 10/6,19,23,25,23,37,51,65ac,71a,79ab
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{\it Sample Exam Questions}
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{\it Disclaimer:} A reasonable 1-2 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).  But you should be prepared for all the possibilities.
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.
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Comments:  Any general formula we have studied can be
used without comment (i.e. you don't need to rederive it in 
you solution).
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I.  Let $Y_1,\ldots,Y_n$ be a random sample from a population
with density function
$$f(y|\theta) = \cases{{3y^2\over\theta^3} & if $0\le y\le \theta$\cr
                            0 & otherwise\cr}$$
containing the parameter $\theta > 0$.
\item{A)} Determine the method of moments estimator for
$\theta$.
\item{B)} Do the method of moments estimator(s) form a consistent
family as $n\to \infty$? 
\item{C)} Determine the maximum likelihood estimator for 
$\theta$.
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II.   Two random samples of women, one from Boston with $n_1 = 300$, one from Dallas 
with $n_2 = 325$ are asked whether they agree with the statement
that all men are basically selfish and egotistical.  
$Y_1 = $ number in the Boston sample who agree, $Y_2 = $ number in
the Dallas. 
\item{A)} State the null and alternative hypotheses, test statistic
and rejection region for a test of the assertion that more
women in Boston feel this way than in Dallas.  Use
$\alpha = .01$.  
\item{B)} The actual data gave $Y_1 = 178$ and $Y_2 = 160$.  
What conclusion do you draw from your test in part A with this data?
\item{C)}  Suppose now we want to test $H_0 : p_1 = .6$ (the Boston
proportion {\it only}) against the alternative $H_0 : p_1 > .6$, using
$\alpha = .01$.
How large would we have to take $n_1$ to get $\beta$ (the 
Type II error probability) $= .05$, if the actual value of $p_1 = .75$?
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III.  Let $X$ be the Brinell hardness measurement of an iron
bar, and assume $X$ is normally distributed with mean $\mu$
and variance $\sigma^2$.  A random sample of 7 bars 
is measured to determine whether the average
strength meets the desired figure $\mu_0 = 172$. 
\item{A)}  The following measurements:
$167, 174, 179, 164, 163, 160, 168$ were obtained.
Test whether the data indicate $\mu < 172$ using $\alpha = .05$
and state your conclusion.
\item{B)}  Test the null hypothesis $H_0 : \sigma^2 = 40$
against the alternative hypothesis $H_a : \sigma^2 \ne 40$
using an appropriate test with $\alpha = .05$, and state your conclusion.
(Hint:  Think about what we did earlier in the course with confidence intervals
for variances.  What is the distribution of ${6 S^2\over \sigma^2}$?  Also see
Section 10.9 if you don't see how to set up a test of this type.)
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IV.  {\it ``Thought Questions'' -- Answer in 2 or 3 complete sentences:}
\item{A)} Exactly why is it the case that
a $Z$-test is not appropriate
when the sample size is small and population variance is not
known?  Explain using the case of a test on a single mean.
\item{B)}  Suppose you do two different tests:
an upper-tail test and a two-tail test.  How would the 
$p$-values of those tests compare if you use the same data for each?  
\item{C)}  If you have several different possible estimators to use
for some parameter in a pdf, how would you make a determination which
one to use?
\item{D)}  Suppose in a statistical test 
the test statistic evaluates to a value that does
not lie in the rejection region, and we ``accept $H_0$.''
Are we claiming that $H_0$ is definitely true in that case? Explain.
\bye
