\magnification=\magstep1
\centerline{Mathematics 376 -- Mathematical Statistics}
\centerline{Final Examination}
\centerline{May 16, 2012}
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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}. 
Do problems I - VI.   Be sure to read each question
carefully before starting to work.  There are 200 possible points,
and up to 20 possible Extra Credit points.
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\noindent
General Note:  A ``random sample'' always consists of independent
measurements from the indicated distribution.
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\noindent
I.  Let $X_1,X_2,X_3,X_4,X_5$ be a random sample from a 
normal distribution with mean $\mu = 6$ and variance $\sigma^2 = 81$.
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\item{A)} (10) What is the distribution (give the type and 
all relevant parameter values) of 
$$U = {(X_1 - 6)^2 + (X_2 - 6)^2 + (X_3 - 6)^2 + (X_4 - 6)^2 \over 81}?$$
Why?
\item{B)} (10)  What is the distribution (give the type and 
all relevant parameter values) of 
$$V = {2(X_5 - 6) \over 9 \sqrt{U}},$$
where $U$ is as in part B?  Why?
\item{C)} (10) What is the distribution (give the type and 
all relevant parameter values) of 
$$W = {(X_1 - 6)^2\over (X_2 - 6)^2}?$$
Why? 
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\noindent
II.  A random variable $Y$ has pdf of the form
$$f(y) = \cases{
{2 y\over \alpha} e^{-y^2/\alpha} & if $y \ge 0$\cr
0 & otherwise\cr}$$
for some unknown $\alpha > 0$.
(This is a particular case of a {\it Weibull} distribution;
these are often used to model length of life of mechanical
components and living organisms.)  Let $Y_1,\ldots,Y_n$ be a 
random sample from this distribution.
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\item{A)} (20)  Find the maximum-likelihood estimator for $\alpha$.  
\item{B)} (15) What is the pdf for the sample maximum, $Y_{(n)}$?   
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\noindent
III.   Let $p$ be the proportion of letters mailed in Belgium
that are delivered the next day. 
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\item{A)} (20)  A random sample of $n = 400$ letters is selected for
tracking and $373$ of those are delivered the next day.  Find an approximate 98\% confidence
interval for $p$ based on this sample.   Is $p = .85$ a ``reasonable''
guess based on this data?
\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 $n = 400$)?  Why does the method you used in part A give a reasonable 
interval estimate for $p$?  Explain.
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\noindent
IV.  The fill weights of a random sample of $n_1 = 13$
boxes of Twigs and Nuts Extra Crunchy Granola produced at Plant 1
had a mean of $20.5$ ounces and standard deviation 
$s = .1$ ounce.  A similar sample of size $n_2 = 11$ 
produced at Plant 2 had mean fill weight $20.3$ ounces and  
standard deviation $s = .083$ ounce.
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\item{A)} (20)  Is there sufficient evidence at the $\alpha = .01$
level to conclude that the variances of 
the fill weights at the two plants
are different?
\item{B)} (20)  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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\noindent
V. The following table gives measurements of water depth
remaining in an evaporating reservoir as a function of time:
$$\matrix{
x {\rm\ time\ (weeks)} & y {\rm\ depth\ (meters) }\cr
      1              &       19.8\cr
      4              &       16.5\cr
      14             &       12.8\cr
      32             &       8.1\cr
      52             &       7.5\cr}$$
\item{A)}  (20)  Find the least-squares estimators for the coefficients
$\beta_0, \beta_1$ in a model $Y = \beta_0 + \beta_1 x + \varepsilon$ for this data set.
\item{B)}  (20)  Is there sufficient evidence to say
that $\beta_1 < -0.1$?  Test with $\alpha = .05$.  
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\noindent
Note:  You may ``purchase'' a printout of an {\tt R} session 
with some results relevant for problem V for {\it 15 points} if you wish.   
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\noindent
VI.  (20) Suppose we have measurements taken from a normal population with
unknown $\mu$ and known $\sigma^2$.  We test $H_0 : \mu = \mu_0$
versus $H_a : \mu > \mu_0$ using a $Z$-test 
and a rejection region designed to produce a Type I error
probability $\alpha$.   Show that in order to make the Type II
error probability equal a given $\beta$ when $\mu$ is equal to some $\mu_a > \mu_0$, 
the sample size should be selected as follows:
$$n = {(z_\alpha + z_\beta)^2 \sigma^2 \over (\mu_a - \mu_0)^2}.$$
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{\it Extra Credit} 
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A New York Times/CBS poll conducted between April 5 and April 12, 2010 
included the question ``Do you approve or disapprove of the way your 
Representative in Congress is handling his or her job?''  The poll was 
carried out in two stages.  First a large national 
simple random sample was asked the question.  The results were reported
in three categories:
$46\%$ of respondents said they approved, $36\%$ said they disapproved, and
$18\%$ said they had no opinion. 
Then the same question was asked of a separate sample of $n = 881$ people, 
all of whom had identified themselves as supporters of the ``Tea Party'' 
movement.  The responses broke down like this in the same 
three categories:
$$\matrix{ & {\rm Approve} &  {\rm Disapprove}  & {\rm No\ opinion} \cr
{\rm Tea\ Party\ Sample} &   352   &   432   &   97  \cr}$$
(Data is adapted from the New York Times web site.)  It seems from the numbers that 
``Tea Party'' supporters may differ from the general population when it comes to
their opinions concerning their Congressional Representatives.  
But is this a real difference, or could it just be a product of chance
variation in the sampling process?   

We did not discuss this in class, but a standard statistical
method in this case would be to use a $\chi^2$-test.  The test statistic
$$X^2 = \sum_{\rm categories} {\left({\rm observed} - {\rm expected}\right)^2 \over {\rm expected}}$$
has approximately a $\chi^2$ distribution with degrees of freedom one less
than the number of categories.  For a test of $H_0:$ the distributions are
really the same, versus $H_a:$ they are significantly different, 
we would reject $H_0$ if $X^2$ fell in an upper tail rejection region.
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\item{A)} (5)  If the ``Tea Party'' sample did follow the national sample
percentages, how many responses would we expect in each category?
\item{B)} (15)  Carry out the $\chi^2$ test and interpret the results.


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\centerline{\it Have a safe, enjoyable, and productive summer!}
\bye
