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\newcounter{prob}
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\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{.4cm}|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}


\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}

\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}

\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}