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\centerline{Mathematics 376 -- Probability and Statistics II}
\centerline{Midterm Exam 1 -- March 11, 2010}
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{\it Directions:}  
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Do all work in the blue exam booklet.  There
are 100 possible points and 10 Extra Credit points (Please turn over for 
question IV and the Extra Credit problem.)
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I. In a random poll of female students at BigState University, 
390 out of 500 sorority women were in favor of lengthening the 
spring break, while 64 out of 100 non-sorority women were 
in favor.  
\item{A)}  (15)  Give a $99\%$ confidence interval for
the difference of the proportions favoring lengthening spring break.
\item{B)}  (5)  Now suppose a new poll of equal numbers of women
from both groups is being planned.  How many women should be 
polled so that the estimate of the difference in proportions 
will be correct to within $.05$?  (Be ``conservative.'')      
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II.  Two methods for teaching reading were applied to two randomly
selected groups of school children, and compared by means of 
a common test.  The sample means ($\overline{y}$) and variances
($s^2$) of the test scores for the two groups are as follows:
$$\matrix{{\rm Statistic} & {\rm Method\ 1} & {\rm Method\ 2}\cr
          {\rm Number\ of\ Children} & 11 & 14 \cr
          \overline{y} & 64 & 69 \cr
          s^2 & 20 & 15 \cr}$$
\item{A)}  (10)  Find a $95\%$ confidence interval for
$\sigma_1^2$ (the variance of the test scores for students
taught with Method 1).  What assumptions are you making?
\item{B)}  (20)  Find a $95\%$ confidence interval for 
$\mu_1 - \mu_2$ (the difference in the population mean test 
scores with the two methods).  What assumptions are you making?
\item{C)}  (5)  Do your results support the conclusion that
Method 2 is a superior method for teaching reading?  
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III.  (15)  A random sample of size $100$ was taken from a 
normal population with $\sigma^2 = 8$.  A confidence interval
for the population mean was given as $6 \pm .25$.  What
was the confidence coefficient associated with this interval
(that is the $\alpha$ value in the $(1 - \alpha)\times 100\%$)?
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IV.  Let $Y_1,\ldots,Y_n$ denote a random sample of size $n$
from a population with a uniform distribution on the interval
$[0,\theta]$.  Let $Y_{(n)}$ be the sample maximum and 
let $U = {1\over \theta} Y_{(n)}$.
\item{A)}  (10)  Find a constant $c$ so that 
$\widehat{\theta} = cY_{(n)}$
is an unbiased estimator for $\theta$.
\item{B)}  (10)  Show that $U$ has CDF given by 
$$F_U(u) = \cases{0 & if $u < 0$\cr
                  u^n & if $0 \le u \le 1$\cr
                  1 & if $u > 1$.
\item{C)}  (10)  Note that the distribution of $U$ does 
not depend on $\theta$.  Use this information to derive
a $95\%$ upper confidence bound on $\theta$ from the 
samples.  
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{\it Extra Credit.}  
\item{A)}  (5)  (Multiple Choice, and explain)
A survey was conducted to determine what features
users want included in cell phone service.  The results 
said a $95\%$ confidence interval showed that
$73\% \pm 4\%$ of users wanted email services.
What is meant by the $\pm 4\%$?
\itemitem{1)}  As many as $4\%$ of those polled may change 
their minds after the time the survey was conducted.
\itemitem{2)}  The probability is $.95$ that the true 
population percentage favoring email service is
in the interval $(69,77)$.
\itemitem{3)}  It would be unlikely to get the observed
sample proportion if the true population percentage
was outside the interval $(69,77)$.  
\itemitem{4)}  Only $4\%$ of cellphone users were polled.
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\item{B)}  Let $Z_1,\ldots,Z_5$ be independent samples
from a $N(0,1)$ population. What is the distribution of
$$W = {4 Z_1^2\over Z_2^2 + \cdots + Z_4^2}?$$
Explain.
\bye
