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\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Review Sheet -- Exam I}
\centerline{February 25, 2010}
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{\it General Information}
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As announced in the course syllabus, the first midterm exam this semester
will be given {\it Thursday, March 11} (the week after Spring Break).  
The exact scheduling will be determined in class on February 25.   
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, and all of Chapter 8).  
There is a more detailed breakdown of topics given below.
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{\it Format}
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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 ``cheat sheet'' for {\it every} formula 
you need or examples, etc. you {\it may not be able to finish the 
exam in the allotted time}.
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{\it Topics}
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The topics to be covered are:
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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 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.  
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{\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.
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{\it Review Session}
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We will review in class on Tuesday, March 9.
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{\it Suggested Review Problems}
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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
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{\it Sample Exam}
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Disclaimer:  The following problems represent the approximate
level of difficulty of the upcoming exam; the actual exam
problems may differ substantially from these.
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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 $Z_1,Z_2,\ldots,Z_7$ 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 
$V = {Z_7 \over \sqrt{U/6}}$.  Explain.  
\item{D)} For which value of $y$ is $P(V \ge y) = .05$?
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II.  A farm grows grapes for jelly.  The following data are measurements 
of grape sugar levels from 8 random samples:
$$16.0, 15.2, 12.0, 16.9, 14.4, 16.3, 15.6, 12.9$$
Assume that these are independent samples from a normal distribution with 
parameters $\mu$ and $\sigma^2$.
\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.
\item{C)}  Construct a $95\%$ confidence interval for 
the population variance $\sigma^2$.
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III.  Two machine shops manufacture toggle levers.  
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.
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IV. Let $Y_1,\ldots,Y_n$ ($n$ fixed but arbitrary)
be random samples from a uniform distribution
on $[0,\theta]$.  Let $\widehat{\theta} = Y_{(1)}$ (the sample
minimum).  Find a constant $c$ so that $c\widehat{\theta}$ 
is an unbiased estimator for $\theta$.
\bye
