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

\begin{center}
{\bf Math 214: Introduction to Statistics}
\\
{\bf Spring Semester 1999}
\\
{\bf Exam~3}
\\
{\bf April~21, 1999}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination consists of 
14  multiple choice problems worth 7 points each. 
No partial credit will be given. 
You start with 2~points.
Record your answers  by placing an $\times$ through one letter for each 
problem.
\\
This booklet consists of 7 sheets of paper including the front cover 
and one blank page at the end.
You are also provided with 3 tables. 
Calculators, books, and notes are not allowed.
\\[1.5ex]
\hline
\end{tabular}
\vspace{.7cm}

\begin{center} 
{\bf Answers to Multiple Choice Problems}
\\[1.3ex]
\begin{minipage}[t]{6cm}
{\bf 1.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 2.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 3.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 4.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 5.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 6.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 7.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) 
\end{minipage}\hspace{2.5cm} \begin{minipage}[t]{6cm}
{\bf \mbox{}\phantom{1}8.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf \mbox{}\phantom{1}9.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 10.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 11.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 12.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 13.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 14.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
\end{minipage}

\vspace*{.4cm}
\fbox{\fbox{
\phantom{M}\begin{minipage}{13cm}
I have not violated the Honor Code in this examination.\rule[-.5cm]{0cm}{1cm}
\\
Signature: \hrulefill
\rule[-.5cm]{0cm}{1.2cm}
\end{minipage}\phantom{M}
}}
\vspace{.7cm}

{\bf Score:} \underline{\hspace{2cm}}
\end{center}


\newpage
The following table is to be used for the questions 1) -- 6). 
\begin{center}
\begin{tabular}{p{.6cm}p{.6cm}|p{1cm}p{1cm}p{.7cm}|p{.7cm}}
     &         & \multicolumn{3}{|c|}{$X_1$}&    \\ 
    &          &  \ 1  &  \ 2     & \ 3      &    \\ \hline
    & \  0   &   0    & 0.05    &    0.1   &    \\ 
 $X_2$& \  2&0.02 & 0.05  & 0.2 &   \\ 
   &  \  4    & 0.08 & 0.1    &   0.4    &     \\  \hline
   &         &         &        &         &   %\rule[-.3cm]{0cm}{1cm}
 \end{tabular}
\end{center}

\begin{problems}
\item
Find $E(X_1)$. 

\begin{tabular}{p{3cm}p{3cm}p{3cm}p{3cm}p{3cm}}
{\bf (a)} $2 $ &
{\bf (b)} $2.86 $ &
{\bf (c)} $0.4 $ &
{\bf (d)} $1 $ &
{\bf (e)} $2.6 $    %correct
\end{tabular}
\vspace*{2.5cm}

\item
Find $P(X_2=2)$.

\begin{tabular}{p{3cm}p{3cm}p{3cm}p{3cm}p{3cm}}
{\bf (a)} $0.2 $ &
{\bf (b)} $0.05 $ &
{\bf (c)} $0.27 $ &   %correct
{\bf (d)} $0.02 $ &
{\bf (e)} $0.1 $ 
\end{tabular}
\vspace*{2.5cm}

\item
Find $P(X_1=2\,|\,X_2=2)$. 

\begin{tabular}{p{3cm}p{3cm}p{3cm}p{3cm}p{3cm}}
{\bf (a)} ${\Ds \frac{1}{4}} $ &
{\bf (b)} ${\Ds \frac{5}{27}} $ &    %correct
{\bf (c)} ${\Ds \frac{2}{27}} $ &
{\bf (d)} ${\Ds \frac{1}{20}} $ &
{\bf (e)} ${\Ds \frac{10}{27}} $  
\end{tabular}
\vspace*{2.5cm}

\item 
Find $P(X_2=4\,|\,X_1\geq 2)$. 

\begin{tabular}{p{3cm}p{3cm}p{3cm}p{3cm}p{3cm}}
{\bf (a)} ${\Ds \frac{0.58}{0.9}}  $  & 
{\bf (b)} ${\Ds \frac{5}{9}}  $ &   %correct
{\bf (c)} ${\Ds \frac{0.5}{0.58}} $   &   
{\bf (d)} ${\Ds \frac{1}{2}}  $ &
{\bf (e)} ${\Ds \frac{0.58}{0.2}}  $  
\end{tabular}
\end{problems}

\newpage
This is the same table again. 
\begin{center}
\begin{tabular}{p{.6cm}p{.6cm}|p{1cm}p{1cm}p{.7cm}|p{.7cm}}
     &         & \multicolumn{3}{|c|}{$X_1$}&    \\ 
    &          &  \ 1  &  \ 2     & \ 3      &    \\ \hline
    & \  0   &   0    & 0.05    &    0.1   &    \\ 
 $X_2$& \  2&0.02 & 0.05  & 0.2 &   \\ 
   &  \  4    & 0.08 & 0.1    &   0.4    &     \\  \hline
   &         &         &        &         &   %\rule[-.3cm]{0cm}{1cm}
 \end{tabular}
\end{center}

\begin{problems}
\setcounter{prob}{4}
\item
Find $E(X_1X_2)$. 

\begin{tabular}{p{3cm}p{3cm}p{3cm}p{3cm}p{3cm}}
{\bf (a)} $7.36 $  & %correct
{\bf (b)} $7.1 $ &
{\bf (c)} $7.54 $   &   
{\bf (d)} $5.45 $ &
{\bf (e)} $6.2 $  
\end{tabular}
\vspace*{2.5cm}


\item
Find $P(X_1+X_2=5)$.

\begin{tabular}{p{3cm}p{3cm}p{3cm}p{3cm}p{3cm}}
{\bf (a)} $0.4 $  & 
{\bf (b)} $0.15 $ &
{\bf (c)} $0.25 $   &   
{\bf (d)} $0.18 $ &
{\bf (e)} $0.28 $       %correct
\end{tabular}
\vspace*{3.5cm}

\junk{
\item
Let $X,\,Y$ be random variables with $E(XY)=7,\,E(X)=2$, and $E(Y)=4$. 
What is the covariance $\cov(X,Y)$ of $X$ and $Y$? 

\begin{choices}
\item $1 $
\item $-1 $   %correct
\item $39 $  
\item $-15 $
\item $0 $   
\end{choices}
}

\item
Let $X,\,Y$ be random variables with variances $V(X)=4,\, V(Y)=9$, and covariance
$\cov(X,Y)=-2$. 
Then the correlation coefficient $\rho$ between $X$ and $Y$ is 

\begin{choices}
\item ${\Ds -\frac{1}{18}} $
\item ${\Ds -\frac{2}{\sqrt{3}}} $
\item ${\Ds -\frac{2}{\sqrt{13}}} $  
\item ${\Ds -\frac{2}{13}} $
\item ${\Ds -\frac{1}{3}} $    %correct
\end{choices}

\newpage
\item
Assume $X$ and $Y$ are independent discrete random variables with 
$P(X=5)=0.3$ and $P(Y=3)=0.5$. 
What is $P(X=5\,|\,Y=3)$? 

\begin{choices}
\item $0.6 $
\item $0.5 $
\item $0.3 $  %correct
\item $0.8 $
\item $0.2 $    
\end{choices}
\vspace*{.5cm}

\item
The weights of 4 miniature poodles (in kilograms)  were found to be
7, 10, 9, and  6. 
The sample mean of these data is 8.
Find the sample variance.

\begin{choices}
\item ${\Ds \sqrt{\frac{10}{3}}}$
\item $10 $
\item ${\Ds \frac{\sqrt{266}}{3}}$  
\item ${\Ds \frac{10}{3}} $  %correct
\item ${\Ds \frac{266}{3}} $    
\end{choices}
\vspace*{.5cm}
\junk{
\item
The distribution of heights of a certain breed of poodles has a mean height of 28 cm 
and a standard deviation of 5 cm. 
Assuming that the sample mean can be measured to any degree of accuracy, find
the probability that the sample mean for a random sample of 36 poodles exceeds
30 cm. 

\begin{choices}
\item $0.3446 $
\item $0.0082 $  %correct
\item $0.4918 $  
\item $0.9918 $
\item $0.1554 $    
\end{choices}
\vspace*{.6cm}
}

\item
Statistics released by the National Highway Traffic Safety Administration show 
that on an average weekend night, 1 out of every 10 drivers on the road is drunk. 
If 400 are randomly checked next Saturday night, what is the approximate 
probability that the number of drunk drivers will be at most~50. 

\begin{choices}
\item $0.4599 $
\item $0.4525 $  
\item $0.0475 $
\item $0.0401 $    
\item $0.9599 $  %correct
\end{choices}

\newpage
\item
Find the probability that a random sample of 15 observations, taken from a normal
population with variance $\sigma^2=2$, will have sample variance greater than 3.
That is, find $P(S^2\geq3)$. 

\begin{choices}
\item $0.99 $
\item $0.01 $
\item $0.05 $  
\item $0.1 $   %correct
\item $0.9 $    
\end{choices}
\vspace*{.6cm}

\item
A random sample of 8 cereal bars of a certain brand was tested for their saturated
fat content.
The sample mean was found to be 0.5 gram and the sample variance was 0.04 
gram.
Give a confidence interval for the population mean $\mu$ with confidence 
coefficient  0.9. 

\begin{choices}
\item $\big(0.5 \pm 1.86\cdot\frac{0.2}{\sqrt{8}}\big)  $
\item $\big(0.5 \pm 1.645\cdot\frac{0.2}{\sqrt{8}}\big)  $  
\item $\big(0.5 \pm 1.895\cdot\frac{0.2}{\sqrt{8}}\big) $  %correct
\item $\big(0.5 \pm 0.9\cdot\frac{0.04}{\sqrt{8}}\big)  $
\item $\big(0.5 \pm 1.895\cdot\frac{0.04}{8}\big)  $    
\end{choices}
\vspace*{.6cm}

\item
A random sample of 100 chocolate energy bars of a certain brand has, on 
average, 230 calories with a standard deviation of 15 calories. 
Construct a confidence interval for the true mean calorie content with confidence 
coefficient 0.95.

\begin{choices}
\item $\big(230 \pm 1.96\cdot\frac{3}{2}\big) $   %correct
\item $\big(230 \pm 0.06\cdot\frac{3}{2}\big) $
\item $\big(230 \pm 1.96\cdot\frac{3}{20}\big) $  
\item $\big(230 \pm 0.13\cdot\frac{3}{2}\big) $
\item $\big(230 \pm 0.95\cdot\frac{3}{2}\big) $    
\end{choices}

\newpage
\item
The scores on a placement test given to college freshmen for the past five years 
are approximately normally distributed with a mean  $\mu=74$ and a 
variance $\sigma^2=36$. 
Find the probability that the sample mean of a random sample of 16 freshmen is
at most 77. 

\begin{choices}
\item $0.4772 $
\item $0.9772 $  %correct
\item $0.0228 $  
\item $1 $
\item $0.1915 $    
\end{choices}

\end{problems}
\end{document}
