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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~1}
\\
{\bf February~8, 1999}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination consists of 
16  multiple choice problems worth 6 points each.
You start with 4~points.
Record your answers  by placing an $\times$ through one letter for each problem
on this answer sheet.
\\
This booklet consists of 8 sheets of paper including the front cover 
and one blank page at the end.
\\
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) \\[1ex]
{\bf 8.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) 
\end{minipage}\hspace{2.5cm} \begin{minipage}[t]{6cm}
{\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]
{\bf 15.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 16.}\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
\begin{problems}
%1
\item
How many 4-digit numbers can be formed from the digits 2, 4, 5, 6, 8, and 9 if each 
digit  can be used only once? 

\begin{choices}
\item $6!$
\item $6^4$
\item $P^6_4$  %correct
\item $4!$
\item $24$
\end{choices}

%13
\item
Assume $A$ and $B$ are events, i.~e. subsets of a sample space $S$.
Let $P$ be the probability defined on $S$. 
It is known that $P(A)=0.6,\, P(B)=0.5$, and $P(A|B)=0.8$.
Calculate $P(AB)$. 

\begin{choices}
\item $0.03$
\item $0.3$
\item $0.1$
\item $0.5$
\item $0.4$  %correct
\end{choices}

%4
\item
How many ways are there to select 3 candidates from 8 equally qualified recent
graduates for 3 identical openings in an accounting firm?

\begin{choices}
\item $P^8_3$
\item $\choose{8}{3}$ %correct
\item $8^3$
\item $P^3_8$
\item $3^8$
\end{choices}

\newpage
%5
\item
Groucho has 6 pairs of socks: blue, yellow,  green, red,  white, and purple. 
These 12 socks are all mixed together in a drawer, and 3 are selected randomly.
What is the probability that 2 of the 3 selected socks match? 

\begin{choices}
\item $\dfrac{3}{12}$
\item $\dfrac{6\cdot10}{\choose{12}{3}}$           %correct
\item $\dfrac{3}{\choose{12}{3}}$
\item $\dfrac{\choose{12}{2}}{\choose{12}{3}}$
\item $\dfrac{6}{P^{12}_3}$
\end{choices}

%6
\item
Maria has in her pocket 3 quarters and 10 nickels.
She reaches into her pocket and selects randomly 8 coins.
What is the probability that the coins she has pulled out add up to 60 cents?

\begin{choices}
\item $\dfrac{\choose{3}{1}\cdot\choose{10}{7}}{\choose{13}{8}}$     %correct
\item $\dfrac{\choose{3}{2}\cdot\choose{10}{2}}{\choose{13}{8}}$
\item $\dfrac{\choose{3}{2}\cdot\choose{10}{2}}{\choose{13}{4}}$
\item $\dfrac{\choose{3}{1}\cdot\choose{10}{7}}{\choose{13}{8}}+
           \dfrac{\choose{3}{2}\cdot\choose{10}{2}}{\choose{13}{4}}$
\item $\dfrac{3}{13}\cdot\dfrac{10}{12}\cdot\dfrac{9}{11}\cdot\dfrac{8}{10}
            \cdot\dfrac{7}{9}\cdot\dfrac{6}{8}\cdot\dfrac{5}{7}\cdot\dfrac{4}{6}$
\end{choices}

\newpage
%7
\item
How many ways can 5  (identical) doughnuts be given to 12 persons, if each 
person is allowed to receive several doughnuts. 

\begin{choices}
\item $P^{12}_5$
\item $\dfrac{12!}{5!}$
\item $5^{12}$
\item $\choose{12}{5}$
\item $12^5$                   %correct
\end{choices}

%8
\item
From  statistics about family summer vacations with a camper it is known that 
in a typical summer 25\% experience mechanical problems, 30\% will receive a 
ticket for committing a traffic violation, and 60\% will have neither of these mishaps. 
What is the percentage of families experiencing both kinds of trouble? 

\begin{choices}
\item 10\%
\item 15\%      %correct
\item 0\%
\item 5\%
\item 20\%
\end{choices}

%9
\item
Suppose that at a certain college 60\% of the students are male. 
It is known that 25\% of the male students smoke, while 30\% of the female 
students smoke. 
What is the fraction of smoking students at that college?

\begin{choices}
\item $55\%$
\item $28\%$
\item $29\%$
\item $27\%$  %correct
\item $26\%$
\end{choices}

\newpage
%10
\item
A bike-store sells mountain bikes and racing bikes.
Based on long-range sales, 
70\% of purchasing customers buy mountain bikes while 30\% buy racing bikes.
Of those that buy a mountain bike, 40\% also buy a helmet. 
But 60\% of racing bike buyers also purchase a helmet. 
A randomly selected customer buys a helmet and a bike. 
What is the probability that it is mountain bike?

\begin{choices}
\item $0.46$
\item $0.7$
\item $\dfrac{0.28}{0.46}$  %correct
\item $\dfrac{0.28}{0.7}$
\item $0.28$
\end{choices}

%11
\item
A discrete random variable $X$ has the following probability distribution
\begin{tabular}{c|cccc}
    $X$   & -2              &        0        &    1          & 3 \\ \hline    
  $p(x)$ & $\frac{1}{4}$ &$\frac{1}{8}$ &$\frac{3}{8}$&$\frac{1}{4}$ 
\end{tabular}
\\
Find the expected value $E(X)$.

\begin{choices}
\item $\dfrac{13}{8}$
\item $\dfrac{3}{4}$
\item $\dfrac{29}{8}$
\item $2$
\item $\dfrac{5}{8}$     %correct
\end{choices}

%15
\item
A random variable $X$ has expected value $E(X)=5$ and variance $V(X)=2$.
What is $V(4X-3)$?

\begin{choices}
\item $8 $
\item $29 $
\item $32 $   %correct
\item $5 $
\item $80 $
\end{choices}

\newpage
%12
\item
A discrete random variable $Y$ has probability distribution 
 \begin{tabular}{c|ccc}
    $Y$   & -1             &        0        &    3        \\ \hline    
  $p(y)$ & $\frac{2}{5}$ &$\frac{1}{5}$ &$\frac{2}{5}$
\end{tabular}\,. 
The expected value is $E(Y)=\dfrac{4}{5}$. 
Find the standard deviation $\sigma(Y)$. 

\begin{choices}
\item $\dfrac{84}{25}$
\item $4$
\item $2$
\item ${\Ds\sqrt{\dfrac{84}{25}}}$   %correct
\item ${\Ds\sqrt{\dfrac{64}{25}}}$
\end{choices}

%2
\item
The set $\{0,\,1,\,\ldots,9\}$ is used to build 9-digit social security numbers. 
Repetition of digits is allowed. 
Assume that all digits are equally likely when creating the numbers. 
What is the probability that a randomly selected SSN has all digits equal 
(like 555555555), but is not the number 000000000\,? 

\begin{choices}
\item $\dfrac{9}{10^9}$          %correct
\item $\dfrac{9}{10!}$
\item $\dfrac{\choose{10}{9}}{10^9}$
\item $\dfrac{1}{10^9}$
\item $\dfrac{9\cdot\choose{10}{9}}{10^9}$
\end{choices}

\newpage
%3
\item
A CBS News/New York Times Poll, taken on February 1, 1999, shows that
56\% of those questioned disapprove of the way the Senate is handling the 
impeachment trial against President Clinton. 
Assume that this poll is representative for the adult US-population. 
For 4 randomly selected adults in the US, let $X$ denote the number of those
disapproving of the Senate's way of handling the impeachment trial.
What is $P(X=1)$? 

\begin{choices}
\item $(0.56)(0.44)^3$
\item $\choose{4}{1}(0.56)^3(0.44)$
\item $\choose{4}{1}(0.56)(0.44)^3$  %correct
\item $P^4_1(0.56)(0.44)^3$
\item $\dfrac{(0.56)(0.44)^3}{4}$
\end{choices}

%14
\item
Let $A$ and $B$ be two independent events having probabilities $P(A)=0.3$ 
and $P(B)=0.4$.
What is $P(AB)$ ?

\begin{choices}
\item $0 $
\item $0.7 $
\item $1.2 $
\item $0.12 $  %correct
\item $0.3 $
\end{choices}

%16
\item
A local company manufactures telephone wire. 
The average length of the wire is 52 inches with a standard deviation of 
6.5 inches. 
At least, what percentage of the telephone wire from this company exceeds 
39 inches?

\begin{choices}
\item $70\% $
\item $75\% $   %correct
\item $80\% $
\item $85\% $
\item $90\% $
\end{choices}

\end{problems}
\end{document}
