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\centerline{Mathematics 376 -- Probability and Statistics 2}
\centerline{Final Examination}
\centerline{May 6, 2006}
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\noindent
{\it Directions}
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Do all work in the blue exam booklet, and {\it include
all work necessary to justify your answers}.  Be sure to read each question
carefully before starting to work.  There are 200 regular
points and 10 Extra Credit points.
{\it General} Note:  A ``random sample'' always consists of {\it independent}
measurements from the indicated distribution.
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\noindent
I.  Let $Y_1,\ldots,Y_n$ be a random sample from a distribution 
with probability density function
$f(y|\theta) = \theta y^{\theta-1}$ if $0 < y < 1$ and $0$ otherwise.
We also assume $\theta > 0$.  
\item{A)} (15) Find the method of moments estimator for $\theta$.
\item{B)} (15) Find the maximum-likelihood estimator for $\theta$. 
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\noindent
II.  Let $X_1,\ldots,X_n$ and $Y_1,\ldots,Y_m$ be two
random samples from normal distributions with common variance $\sigma^2$.
\item{A)} (10) Using the fact that the expected value of a $\chi^2$
random variable with $\nu$ degrees of freedom is $\nu$, 
show that $S_1^2 = {1\over n-1} \sum_{i=1}^n (X_i - \overline X)^2$
is an unbiased estimator for $\sigma^2$.  
\item{B)} (10) Show that the pooled estimator 
$S_p^2$ using both the $X_i$ and the $Y_j$ is also unbiased for $\sigma^2$.  
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\noindent 
III. 
\item{A)} (15) Let $Y_1,Y_2,\ldots,Y_{10}$ be a random sample from a 
normal distribution with mean $\mu = 2$ and variance $\sigma^2 = 81$.
Let $U = {1\over 81}\sum_{i=1}^9 (Y_i - 2)^2$.  What is the distribution of 
$V = {Y_{10} - 2\over 3\sqrt{U}}$?
\item{B)} (15)  If $T$ has a $t$-distribution with $\nu$ degrees
of freedom, what is the distribution of $T^2$?  Explain.
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\noindent
IV. Let $p$ be the proportion of letters mailed in the Netherlands
that are delivered the next day.
\item{A)} (15)  A random sample of $n = 200$ letters are sent
out and $142$ are delivered the next day.  Find an approximate 95\% confidence
interval for $p$ based on this sample. 
\item{B)} (15)  (``Thought question'')  Note that part A says
``approximate.''  What is the actual distribution of $Y = $
the number of letters delivered the next day (out of a random sample of 
size $n = 200$)?  Why does the method you used in part A give a reasonable 
interval estimate for $p$?
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\noindent
V.  A mathematics department wishes to evaluate a new
method of teaching calculus with Maple labs.  At the end
of the course, $15$ students who used the labs are given a 
standardized test.  Their average score is $83$, with standard 
deviation $9$.
\item{A)} (10) Find a $95\%$ confidence interval for the 
mean test score for students who are taught using the new method.  
\item{B)} (10) From departmental experience, students
who are taught the course without the Maple labs average 79
on the same standardized test, and the standard deviation
is also 9 for these students.  Is there sufficient evidence
to conclude that taking the course with the
labs has an effect on students' performance, at the $\alpha = .05$
level?
\item{C)} (10) What number $n$ of students who took the course
with the labs would have to be tested in order for
the department to be ``98\% sure'' that the sample mean 
test score is no farther than 1 away from the true population mean 
test score for the students taking the course with the labs.
You may assume for this part that $n$ will be significantly $ > 30$. 
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VI.  The fill weights of a random sample of $n_1 = 21$
6-pound boxes of ``Super-Sudsy''
laundry soap produced at Plant 1
had a mean of $6.25$ pounds and standard deviation 
$s = .095$ pounds.  A similar sample of size $n_2 = 21$ 
produced at Plant 2 had mean weight $6.12$ pounds and  
standard deviation $s = .065$.
\item{A)} (15)  Is there sufficient evidence at the $\alpha = .01$
level to conclude that the standard deviations at the two plants
are different?
\item{B)} (15)  Is there sufficient evidence to conclude
that the mean fill weights are different?
Report the results by giving an estimate of the $p$-value of your test.

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VII.  The following table gives measurements of the firmness
of pickles stored in low-salt brine as a function of time:
$$\matrix{
x {\rm\ time\ (weeks)} & y {\rm\ firmness\ (lb)}\cr
      1              &       19.8\cr
      4              &       16.5\cr
      14             &      12.8\cr
      32             &       8.1\cr
      52             &       7.5\cr}$$
\item{A)}  (15)  Find the least-squares estimators for the coefficients
in a model $Y = \beta_0 + \beta_1 x + \varepsilon$ for this data set.
\item{B)}  (15)  Is there sufficient evidence to say
that $\beta_0 < 21$?  Explain, using the $p$-value of an
appropriate test.  
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{\it Extra Credit}  (Short Essay -- no more than
a paragraph, please!) (10)  Suppose we needed to choose between
the estimators in question I, parts A and B.  One criterion for
comparing estimators we discussed in class used the relative
efficiency ${\rm eff}(\widehat{\theta_1},\widehat{\theta_2}) =
V(\widehat{\theta_2})/V(\widehat{\theta_1})$.  But it shouldn't
be clear how to compute either variance theoretically from 
your formulas.  Propose a method to test which estimator is
superior ``experimentally,'' taking into account the possibility
that which estimator is superior might depend on the true value
of $\theta$.
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\centerline{\it Have a safe, enjoyable, and productive summer!}
\bye