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\newcounter{prob}
\newcounter{subprob}
\newenvironment{problems}{
     \begin{list}{\bf{\arabic{prob}.}\hfill}{\usecounter{prob}
     \topsep-1.4ex \labelwidth.6cm \leftmargin.7cm \labelsep.1cm
     \rightmargin0cm \parsep2ex \itemsep2ex
     \partopsep1.4ex}}{\end{list}}
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     \begin{list}{\bf{(\alph{subprob})}\hfill}{\usecounter{subprob}
     \topsep-1.4ex \labelwidth.6cm \leftmargin.8cm \labelsep.2cm
     \rightmargin0cm \parsep2ex \itemsep2ex
     \partopsep1.4ex}}{\end{list}}
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\begin{document}
\mbox{}\vspace*{-1.5cm}

{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Heide Gluesing-Luerssen\hspace*{.7cm}

\begin{center}
{\bf Mathematics 214: Introduction to Statistics }
\\
{\bf Spring Semester 1998}
\\
{\bf Final Exam}
\\
{\bf May 4, 1998}
\end{center}
%\vspace*{.4cm}
\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.5cm}Examination
contains 30 questions 5 points worth each.
The highest possible score is 150 points.
Fill in your answers on this cover sheet by placing an {\sf X} through one
letter for each problem.
Books and notes are not allowed.
\\[1.3ex]
\hline
\end{tabular}
\vspace*{.5cm}

\mbox{}\hspace*{2.5cm}\begin{tabular}[t]{|p{.5cm}||p{.4cm}|p{.4cm}|p{.4cm}|p{.4c
m}|p{.4cm}||p{.6cm}|}\hline
 \ \ 1      &\  a &\ b &\ c &\ d &\ e &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ \ 2      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ \ 3      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ \ 4     &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ \ 5      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ \ 6      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm}\\ \hline
 \ \ 7      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm}\\ \hline
 \ \ 8      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ \ 9      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 10      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 11      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 12     &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm}\\ \hline
 \ 13      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm}\\ \hline
 \ 14      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 15      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
\end{tabular} \qquad\quad
\begin{tabular}[t]{|p{.5cm}||p{.4cm}|p{.4cm}|p{.4cm}|p{.4cm}|p{.4cm}||p{.6cm}|}\
hline
 \ 16      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 17      &\  a &\ b &\ c &\ d &\ e &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 18     &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 19      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 20     &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm}\\ \hline
 \ 21      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 22      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm}\\ \hline
 \ 23      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 24      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 25      &\  a &\ b &\ c &\ d &\ e  &    \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 26     &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 27      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm} \\ \hline
 \ 28     &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm}\\ \hline
 \ 29     &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm}\\ \hline
 \ 30     &\  a &\ b &\ c &\ d &\ e  &   \rule[-.3cm]{0cm}{0.8cm}\\ \hline
 \end{tabular}
\\[1ex]
\mbox{}\hspace*{13.8cm} {\bf Total} \fbox{\rule[-.3cm]{0cm}{.9cm}\qquad\ \ }
\vspace*{.5cm}

{\bf Sign the pledge:}
\\[1ex]
``On my honor, I have neither given nor received unauthorized aid on this
Exam.''

{\bf Signature:} \underline{\mbox{\hspace*{7cm}}}

%\vspace*{.3cm}
\begin{center}     {\bf GOOD LUCK}\end{center}

\newpage
\begin{problems}
\item
A fair die is tossed 5 times. What is the probability that a ``five'' comes
up exactly 3 times?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{{6\choose1}{6\choose 5}}{6!}}$     &
{\bf b)} & ${\Ds {5\choose 3}\Big(\frac{1}{6}\Big)^3\Big(\frac{5}{6}\Big)^2}$ &
{\bf c)} &  ${\Ds {5\choose
3}\Big(\frac{1}{6}\Big)^2\Big(\frac{5}{6}\Big)^3}$  &
{\bf d)} & ${\Ds {5\choose 3}\Big(\frac{1}{6}\Big)^3}$ &
{\bf e)} & ${\Ds \Big(\frac{1}{6}\Big)^3\Big(\frac{5}{6}\Big)^2}$
\end{tabular}
\vspace*{3.5cm}

\item
Five employees of a firm are ranked from 1 to 5 based on their ability to
program a computer.
Three of these employees are selected to fill three identical programming jobs.
If all possible choices of three (out of five) are equaly likely, what is
the probability that the highest-ranked
employee among those selected has rank 4?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{3}{10}}$     &
{\bf b)} & ${\Ds \frac{1}{5}}$ &
{\bf c)} &  ${\Ds \frac{P^5_2}{5!}}$  &
{\bf d)} & ${\Ds \frac{P^5_3}{5^3}}$ &
{\bf e)} & ${\Ds \frac{2}{5^3}}$
\end{tabular}
\vspace*{3.5cm}


\item
How many 7-digit phone-numbers be constructed using the digits
$0,\,1,\,\ldots,\,9$, if the first digit has to
be nonzero.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 9\cdot10^7}$     &
{\bf b)} & ${\Ds 10^7}$ &
{\bf c)} &  ${\Ds 9^7}$  &
{\bf d)} & ${\Ds 10^7-9^7}$ &
{\bf e)} & ${\Ds 9\cdot10^6}$
\end{tabular}
\vspace*{3.5cm}
\junk{
\item
A box contains two blue, two yellow, and three red marbles.
Selst four marbles randomly. What is the probability that at most 1 is red?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{12}{35}}$     &
{\bf b)} & ${\Ds \frac{3}{7}}$ &
{\bf c)} &  ${\Ds \frac{12}{35^2}}$  &
{\bf d)} & ${\Ds \frac{P^7_1}{P^7_3}}$ &
{\bf e)} & ${\Ds \frac{13}{35}}$
\end{tabular}
\vspace*{3.5cm}
}

\item
A discrete random variable takes the values $-1,\,3$, and $5$ with the
probabilities
$P(X=-1)=0.5,\, P(X=3)=0.3$, and $P(X=5)=0.2$.
Calculate $E(X^2)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 8.2}$     &
{\bf b)} & ${\Ds 0.72}$ &
{\bf c)} &  ${\Ds 7.2}$  &
{\bf d)} & ${\Ds 0.38}$ &
{\bf e)} & ${\Ds 1.4}$
\end{tabular}


\newpage
\item
If 30\% of the persons donating blood at a clinic have O$^+$ blood, find
the probability that
one has to wait for exactly the 10th donor to get the 3rd O$^+$-donation.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds (0.7)^7}$     &
{\bf b)} & ${\Ds {9\choose2}(0.3)^3(0.7)^7}$ &
{\bf c)} &  ${\Ds {10\choose3}(0.3)^3(0.7)^7}$  &
{\bf d)} & ${\Ds {9\choose2}(0.7)^3(0.3)^7}$ &
{\bf e)} & ${\Ds 3(0.3)^3(0.7)^7}$
\end{tabular}
\vspace*{3cm}

\item
You roll a fair die three times in a row. What is the probability that the
sum of the upper face  numbers
is 6?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{3}{216}}$     &
{\bf b)} & ${\Ds \frac{13}{216}}$ &
{\bf c)} &  ${\Ds \frac{10}{216}}$  &
{\bf d)} & ${\Ds \frac{7}{216}}$ &
{\bf e)} & ${\Ds \frac{12}{216}}$
\end{tabular}
\vspace*{3cm}

\item
A blood test for $X$, a certain disease, has the following accuracy:

\mbox{}\hspace*{4cm}\begin{tabular}{p{3cm}||c|c}
      &\multicolumn{2}{|c}{ test result} \\
      & positive & negative \\\hline\hline
 patient with $X$ & 0.99 & 0.01 \\ \hline
  patient without $X$ & 0.02 & 0.98
\end{tabular}

That is, if a person has $X$, then the probability of a positive test
result is 0.99, etc.
The disease rate for $X$ in the general population is 0.001.
If a person tests positive for $X$, what is the probability that the person
has $X$?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.009}$     &
{\bf b)} & ${\Ds 0.09}$ &
{\bf c)} &  ${\Ds 0.25}$  &
{\bf d)} & ${\Ds 0.047}$ &
{\bf e)} & ${\Ds 0.99}$
\end{tabular}
\vspace*{3cm}

\item
Let $S$ be a sample space and $A,\,B$ events associated with $S$.
Assume that $P(A)=0.4,\,P(B)=0.3$, and $P(A\cap B)=0.1$.
Calculate $P(\overline{B}\,|\,A)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.28}$     &
{\bf b)} & ${\Ds \frac{0.28}{0.4}}$ &
{\bf c)} &  ${\Ds \frac{1}{4}}$  &
{\bf d)} & ${\Ds \frac{1}{2}}$ &
{\bf e)} & ${\Ds \frac{3}{4}}$
\end{tabular}

\newpage
\item
Let $X$ be a continuous random variable with probability density function
\[
         f(x)=\left\{\begin{array}{ll}
                            \frac{3}{4}(x^2-1),&1\leq x\leq 2 \\[1ex] 0, &
\mbox{\rm elsewhere}\end{array}\right.
\]
Find $E(X)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{27}{16}}$     &
{\bf b)} & ${\Ds \frac{15}{16}}$ &
{\bf c)} &  ${\Ds \frac{7}{4}}$  &
{\bf d)} & ${\Ds \frac{9}{16}}$ &
{\bf e)} & ${\Ds 1}$
\end{tabular}
\vspace*{4cm}

\item
Let $X$ be a continuous random variable with probability density function
\[
         f(x)=\left\{\begin{array}{ll}
                            \frac{3}{8}x^2,&0\leq x\leq 2 \\[1ex] 0, &
\mbox{\rm elsewhere}\end{array}\right.
\]
Calculate $V(X)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{39}{4}}$     &
{\bf b)} & ${\Ds 0}$ &
{\bf c)} &  ${\Ds \frac{12}{5}}$  &
{\bf d)} & ${\Ds \frac{3}{20}}$ &
{\bf e)} & ${\Ds \frac{9}{10}}$
\end{tabular}
\vspace*{4cm}

\item
Let $X$ be as in problem~10.
Calculate $P(X\geq1)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 1}$     &
{\bf b)} & ${\Ds \frac{7}{8}}$ &
{\bf c)} &  ${\Ds \frac{1}{8}}$  &
{\bf d)} & ${\Ds \frac{29}{32}}$ &
{\bf e)} & ${\Ds \frac{1}{2}}$
\end{tabular}

\newpage
\item
The number of telephone calls coming into a central switchboard of an
office building have a Poisson Distribution
and average 5 per minute.
Find the probability that at least 2 calls will arrive in a given 1-minute
period.

\begin{tabular}{cp{2.2cm}cp{1.5cm}cp{3.8cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds  e^{-5}+5e^{-5}}$     &
{\bf b)} & ${\Ds 2e^{-5}}$ &
{\bf c)} &  ${\Ds 1-e^{-5}-5e^{-5}-\frac{25}{2} e^{-5}}$  &
{\bf d)} & ${\Ds 1-e^{-5}-5e^{-5}}$ &
{\bf e)} & ${\Ds \frac{25}{2} e^{-5}}$
\end{tabular}
\vspace*{3.5cm}

\item
A telephone call arrived at a switchboard at a random time within a
1-minute interval.
The switchboard was fully busy for 15 seconds into this 1-minute period.
Find the probability that the call arrived when the switchboard was not
fully occupied.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds e^{-4}}$     &
{\bf b)} & ${\Ds\frac{3}{4}}$ &
{\bf c)} &  ${\Ds \frac{1}{4}}$  &
{\bf d)} & ${\Ds 1-e^{-4}}$ &
{\bf e)} & ${\Ds e^{-1}}$
\end{tabular}
\vspace*{3.5cm}

\item
The service times at teller windows in a bank were found to follow an
exponential distribution with a mean
of 4 minutes.
A customer arrives at 2:00 p.m.
Find the probability that she will still be there at 2:03 p.m.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 1-e^{-\frac{3}{4}}}$     &
{\bf b)} & ${\Ds \frac{1}{4}e^{-\frac{3}{4}}}$ &
{\bf c)} &  ${\Ds 1-\frac{3}{4}e^{-\frac{3}{4}}}$  &
{\bf d)} & ${\Ds \frac{1}{3}e^{-\frac{1}{4}}}$ &
{\bf e)} & ${\Ds e^{-\frac{3}{4}}}$
\end{tabular}
\vspace*{3.5cm}

\item
Let $X$ be a random variable with a normal probability density function
having mean 5 and variance 9.
Find $P(X\leq8)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.3413}$     &
{\bf b)} & ${\Ds 0.8413}$ &
{\bf c)} &  ${\Ds 0.1587}$  &
{\bf d)} & ${\Ds 0.6293}$ &
{\bf e)} & ${\Ds 0.1293}$
\end{tabular}

\newpage
\item
A fair coin is tossed 2,000 times.  What is the approximate probability
that 1,050 or more ``heads'' come up?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.9864}$     &
{\bf b)} & ${\Ds 0.4864}$ &
{\bf c)} &  ${\Ds 0.0136}$  &
{\bf d)} & ${\Ds 0.0985}$ &
{\bf e)} & ${\Ds 0.4015}$
\end{tabular}
\vspace*{3.5cm}
\end{problems}

The following table is to be used for the questions 17) -- 19).
\begin{center}
\begin{tabular}{p{1cm}|p{1cm}|p{1cm}p{1cm}p{1cm}|p{.7cm}}
             &           & \multicolumn{3}{|c|}{$X$}&    \\ \hline
             &            &  \ -1   & \  0     & \ 2      &    \\ \hline
             & \   0    &  0.2    & 0.1   & 0.05    &   \\
     $Y$   &  \  1    & 0.25   & 0.1    &   0     &     \\
             &  \   2    & 0.15  & 0.1    &  0.05   &    \\ \hline
            &         &         &        &         &   %\rule[-.3cm]{0cm}{1cm}
 \end{tabular}
\end{center}

\begin{problems}
\setcounter{prob}{16}

\item
Find $P(X=2\,|\,Y=0)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.5}$     &
{\bf b)} & ${\Ds \frac{1}{3}}$ &
{\bf c)} &  ${\Ds \frac{1}{7}}$  &
{\bf d)} & ${\Ds 0.05}$ &
{\bf e)} & ${\Ds 0.45}$
\end{tabular}
\vspace*{2.7cm}

\item
Find $E(XY)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds -0.45}$     &
{\bf b)} & ${\Ds -0.35}$ &
{\bf c)} &  ${\Ds 1}$  &
{\bf d)} & ${\Ds 0.25}$ &
{\bf e)} & ${\Ds -0.2}$
\end{tabular}
\vspace*{3cm}

\item
Find $V(X)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.84}$     &
{\bf b)} & ${\Ds 0.6}$ &
{\bf c)} &  ${\Ds 1.16}$  &
{\bf d)} & ${\Ds 1}$ &
{\bf e)} & ${\Ds 1.4}$
\end{tabular}

\newpage
\item
Let $X,\,Y$ be independent random variables with $E(X)=3,\,E(Y)=4$ and
$V(X)=9,\,V(Y)=25$.
Then the covariance ${\sl Cov}(X,Y)$ is

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 3}$     &
{\bf b)} & ${\Ds -3}$ &
{\bf c)} &  ${\Ds 0}$  &
{\bf d)} & ${\Ds 27}$ &
{\bf e)} & ${\Ds 15}$
\end{tabular}
\vspace*{3.5cm}

\item
Let $X,\,Y$ be random variables with $V(X)=9,\,V(Y)=25$, and ${\sl
Cov}(X,Y)=20$.
Find $V(3X-Y)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 22}$     &
{\bf b)} & ${\Ds -18}$ &
{\bf c)} &  ${\Ds -58}$  &
{\bf d)} & ${\Ds 42}$ &
{\bf e)} & ${\Ds -14}$
\end{tabular}
\vspace*{3.5cm}

\item
The service times for customers coming through a certain checkout counter
are independent random variables
with a mean of 2.5 minutes and a variance of 1 minute.
Approximate the probability that 50 customers can be serviced in less than
2 hours.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.5398}$     &
{\bf b)} & ${\Ds 0.2389}$ &
{\bf c)} &  ${\Ds 0.2611}$  &
{\bf d)} & ${\Ds 0.7611}$ &
{\bf e)} & ${\Ds 0.0398}$
\end{tabular}
\vspace*{3.5cm}

\item
For an aptitude test for a certain job history shows scores to be normally
distributed with a variance of 225.
20 applicants are to take the test.
Find the approximate probability that the sample variance of the test
scores will be less than 140.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.1}$     &
{\bf b)} & ${\Ds 0.9}$ &
{\bf c)} &  ${\Ds 0.975}$  &
{\bf d)} & ${\Ds 0.025}$ &
{\bf e)} & ${\Ds 0.05}$
\end{tabular}

\newpage
\item
A random poll of the adults in the U.S.A. has been conducted to see how
many people favor football as a sport.
The result was that 36\% of the 1091 adults sampled list football as their
favorite sport.
Find a 95\% confidence interval for the true fraction of the U.S. adults
who favor football.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds (0.34,\,0.38)}$     &
{\bf b)} & ${\Ds (0.35,\,0.37)}$ &
{\bf c)} &  ${\Ds (0.33,\,0.39)}$  &
{\bf d)} & ${\Ds (0.32,\,0.4)}$ &
{\bf e)} & ${\Ds (0.31,\,0.41)}$
\end{tabular}
\vspace*{4cm}

\item
The temperature of the water coming out of a certain power plant is
specified by law to be less than or equal to
75 degrees Fahrenheit.
The sample mean for a random sample of 16 temperatures taken on a given day
is 77 degrees.
The sample variance is 9.
Assume that the temperature measurements have a normal distribution.
What is the smallest significance level with which we can reject the null
hypothesis that the temperature is less than
$75$?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.1}$     &
{\bf b)} & ${\Ds0.05}$ &
{\bf c)} &  ${\Ds 0.025}$  &
{\bf d)} & ${\Ds 0.01}$ &
{\bf e)} & ${\Ds 0.005}$
\end{tabular}
\vspace*{4.5cm}

\item
Suppose that a random sample of the volumes in cubic feet of 30 black
cherry trees in the Allegheny National Forest
has been taken.
Assume that the sample mean for the volumes of the 30  trees is
$\bar{x}=30.2$ and the sample variance is
$s^2=268.96$.
If we assume further that the distribution of the volumes of all the black
cherry trees is normal, then a confidence
interval for the variance $\sigma^2$ with confidence coefficient $0.9$ is
given by


\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds (184.3,\,436.3)}$     &
{\bf b)} & ${\Ds (178.2,421.8)}$ &
{\bf c)} &  ${\Ds (189.6,\,455.7)}$  &
{\bf d)} & ${\Ds (149.0,\,594.5)}$ &
{\bf e)} & ${\Ds (183.3,\,440.5)}$
\end{tabular}

\newpage
\item
Prospective employees of an engineering firm are told that engineers in the
firm work at least 45 hours per
week,  on the average.
A random sample of 40 engineers in the firm showed that, for a particular
week,  they averaged 44 hours of work
with a standard deviation of 3 hours.
What is the smallest significance level with which we can reject the null
hypothesis that the average working time
per week is at least 45 hours?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.9826}$     &
{\bf b)} & ${\Ds 2.108}$ &
{\bf c)} &  ${\Ds 0.0348}$  &
{\bf d)} & ${\Ds 0.0174}$ &
{\bf e)} & ${\Ds 0.4826}$
\end{tabular}
\vspace*{4cm}

\item
One-hour carbon monoxide concentrations in air samples from a large city
average 10 ppm, with a standard deviation
of 9 ppm.
Find the probability that the average concentration in 100 samples selected
randomly will exceed 12 ppm.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 0.4129}$     &
{\bf b)} & ${\Ds 0.0132}$ &
{\bf c)} &  ${\Ds 0.4868}$  &
{\bf d)} & ${\Ds 0.0871}$ &
{\bf e)} & ${\Ds 0.2514}$
\end{tabular}
\vspace*{3.5cm}

\item
The amount of Guinness sold weekly by McGuires Irish pub is normally
distributed with mean 2,000 pints and
standard deviation of 500 pints.
How many pints should Mr.~McGuire have on hand at the beginning of the week
to be 97.5\% sure that he will not
run out before the end of the week?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 2,044}$     &
{\bf b)} & ${\Ds 1,020}$ &
{\bf c)} &  ${\Ds 2,980}$  &
{\bf d)} & ${\Ds 1956}$ &
{\bf e)} & ${\Ds 4,420}$
\end{tabular}
\vspace*{3.5cm}

\item
Each time a soccer player takes a penalty, her chances of scoring a goal
are 0.7.
Let $Y$ be the number of goals she scores out of 5 penalties.
What is the standard deviation of $Y$?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \sqrt{1.05}}$     &
{\bf b)} & ${\Ds 1.05}$ &
{\bf c)} &  ${\Ds 3.5}$  &
{\bf d)} & ${\Ds\sqrt{3.5}}$ &
{\bf e)} & ${\Ds \sqrt{1.5}}$
\end{tabular}
\vspace*{3.5cm}




\end{problems}
\end{document}
