\magnification=\magstep1
\centerline{Mathematics 376 -- Mathematical Statistics}
\centerline{Review Sheet -- Exam I}
\centerline{February 24, 2012}
\bigskip
\noindent
\bigskip
\noindent
{\it General Information}
\bigskip
As announced in the course syllabus, the first midterm exam this semester
will be given {\it Friday, March 16} (the week after Spring Break), or
possibly in the evening of Thursday, March 15.  
The exact scheduling will be determined in class on Monday, February 27.   
The exam will cover the material we have discussed since the start of 
the semester, up to and including the material on confidence intervals
for variances (this is section 7.2, section 6.7 on the order statistics, 
and all of Chapter 8), plus the material on Lab Project/Problem Set 4.  
There is a more detailed breakdown of topics given 
below.
\bigskip
\noindent
{\it Format}
\bigskip
This will be an {\it closed book exam}, but you may
place any information you think might be useful on 
one side of a standard $8.5 \times 11$ sheet of paper, 
bring that to the exam and consult it at any time. 
I will also provide copies of any tables from the text
you will need to use.  Even though
you will have all that information to use, 
you should still prepare very carefully for the exam and 
understand the key concepts we have talked about.  Know
how to apply the different techniques we have studied.
If you are consulting your sheet for {\it every} formula 
you need or examples, etc. you {\it may not be able to finish the 
exam in the allotted time}.
\bigskip
\noindent
{\it Topics}
\bigskip
\noindent
The topics to be covered are:
\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)} 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. 
Also be able to analyze estimators to determine whether they are biased or 
not, construct unbiased estimators, etc.  (see Problem Set 2).  
\item{3)}  The pdf's for order statistics (especially the 
sample maximum and minimum), and how they can be used for estimation
problems -- this was covered in class; also see 
section 6.7 in the text.
\item{4)}  The concept of a confidence interval (interval estimator),
derivation of formulas for interval estimators via pivotal quantities.
\item{5)}  Large-sample confidence intervals for means, differences of means,
proportions, differences of proportions (constructed using pivotal quantities
that are normal)
\item{6)}  Small-sample confidence intervals for means, differences of means
(constructed using pivotal quantities that have $t$-distributions)
\item{7)}  Confidence intervals for variances, ratios of variances.  
\item{8)}  From Lab Project/Problem Set 4:  Normal quantile-quantile plots
as a way to test for normality.
\bigskip
\noindent
{\it Comment:}  As you can tell, much of this depends heavily on the 
probability topics we learned last semester.  If you are feeling ``rusty''
on general properties of expected values or variances like the 
important relations
$$E(aX+bY) = aE(X) + bE(Y)$$
and 
$$V(aX+bY) = a^2V(X) + b^2V(Y) + 2ab {\rm Cov}(X,Y),$$ 
or on computing expected values or variances of random variables with
given distribution (i.e. given pdf), start by reviewing that material.
\bigskip
\noindent
{\it Review Session}
\bigskip
We will review in class on Wednesday, March 14.
\bigskip
\noindent
{\it Suggested Review Problems}
\bigskip
From the text: 
\item{} Chapter 7/15,20,33,37,38,39,95; 
\item{} Chapter 8/9,15,19,23,43,61,67,83,91,95,103
\bigskip
\noindent
{\it Sample Exam}
\bigskip
\noindent
{\it Disclaimer}:  The following problems represent the approximate
level of difficulty of the upcoming exam; the actual exam
problems may differ substantially from these and these questions are
somewhat longer than the actual exam will be to give an idea of
the range of different sorts of things that might be asked.
\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 $Z_1,Z_2,\ldots,Z_7$ be independent random samples from
a standard normal distribution.  Let 
$$W = Z_1^2 + Z_2^2 + \cdots + Z_7^2$$
\item{A)} What is the distribution of $W$?  Explain.
\item{B)} Find $P(1.68987 \le W \le 14.0671)$.  
\item{C)} Let $U = Z_1^2 + Z_2^2 + \cdots + Z_6^2$
(not including $Z_7$).   What is the distribution of 
$V = {6 Z_7^2 \over U}$.  Explain.  
\item{D)} For which value of $y$ is $P(V \ge y) = .05$?
\item{E)} How would you find the $y$ such that $P(V \ge y) = .95$?
\item{F)} Use the $Z_i$ to construct a random variable $T$ that
has a $t$-distribution with $\nu = 5$ degrees of freedom.  
\bigskip
\noindent
II. Let $Y_1,\ldots,Y_n$ ($n$ fixed but arbitrary)
be random samples from an exponential distribution with pdf
$$f(y) = \cases{{e^{-y/\theta}\over\theta} & if $y > 0$\cr
                     0 & otherwise\cr}$$
\item{A)}  What is $\lim_{n\to\infty} P(Y_{(n)} \ge 2\theta)$?  ($Y_{(n)}$ is the sample maximum, as usual.)  
Is your answer intuitively reasonable, given what we know about exponential distributions?
\item{B)}  Let $\widehat{\theta} = Y_{(1)}$ (the sample
minimum).  Find a constant $c$ so that $c\widehat{\theta}$ 
is an unbiased estimator for $\theta$.
\bigskip
\noindent
III.  A farm grows grapes for jelly.  The following data are measurements 
of grape sugar levels from 8 random samples:
$$16.0, 15.2, 12.0, 16.9, 14.4, 16.3, 15.6, 12.9.$$
\item{A)}  Find point estimates $\mu$ and $\sigma$ from the given information.
\item{B)}  Construct a $90\%$ confidence interval for the population 
mean $\mu$ from the given information.  Explain how you know which method to use,
and state all assumptions you are making for this method to be applicable.  What 
method(s) could you use to test whether these assumptions are reasonable?
\item{C)}  From your answer to part B, can we rule out the possibility that
the value of $\mu$ is $16$?  
\item{D)}  Construct a $95\%$ confidence interval for 
the population variance $\sigma^2$.
\bigskip
\noindent
IV.  Two machine shops manufacture toggle levers (whatever those are!).  
A random sample of size $642$ from the output of shop $A$
produced $24$ defective toggle levers.
A random sample of size $500$ from the output of shop $B$
produced $22$ defective toggle levers. 
\item{A)}  Give a point estimate of $p_A$, the proportion
of defectives in the output of shop A.  
\item{B)}  Derive a $95\%$ confidence interval for $p_A$.
\item{C)}  Derive a $95\%$ confidence interval the difference
$p_A - p_B$ where $p_B$ is the proportion of defectives
in the output of shop B.  Is there good reason to suppose
the output of shop A has higher quality than the output of 
shop B?  Explain.
\item{D)}  How large a sample would be needed from the output
of shop $A$ in order that 
$$P(|\widehat{p_A} - p_A| < .01) \ge .95$$
(i.e. the probability that the estimator is within $.01$ of the true value
is at least $.95$).   

\bye
