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\begin{document}
\mbox{}\vspace*{-1.5cm}

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

\begin{center}
{\bf Mathematics 214: Introduction to Statistics }
\\
{\bf Spring Semester 1998}
\\
{\bf Exam 2}
\\
{\bf March 4, 1998}
\end{center}
%\vspace*{.4cm}
\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.8cm}Examination
contains 16 questions 6 points worth each.
You start with 4 points,  and the highest possible score is 100.
Fill in your answers on this cover sheet by placing an {\sf X} through one
letter for each problem except problems 7 and 8.
Write the answers for 7 and 8 in the corresponding empty boxes of the list
below.
Calculators, books, and notes are not allowed.
\\[1.7ex]
\hline
\end{tabular}
\vspace*{.7cm}

\mbox{}\hspace*{1.5cm}\begin{tabular}[t]{|p{.6cm}||p{.5cm}|p{.5cm}|p{.5cm}|p{.5c
m}|p{.5cm}||p{.8cm}|}\hline
 \ 1      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 2      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 3      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 4     &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 5      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 6      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 7      &\multicolumn{5}{|c||}{} &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 8      &\multicolumn{5}{|c||}{}  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
\end{tabular} \qquad\quad
\begin{tabular}[t]{|p{.6cm}||p{.5cm}|p{.5cm}|p{.5cm}|p{.5cm}|p{.5cm}||p{.8cm}|}\
hline
 \ \ 9      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 10      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 11      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 12     &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 13      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 14      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 15      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \ 16      &\  a &\ b &\ c &\ d &\ e  &   \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 \end{tabular}
\\[1.3ex]
\mbox{}\hspace*{14.3cm} {\bf Total} \fbox{\rule[-.3cm]{0cm}{1cm}\qquad\ }

\vspace*{1.2cm}
{\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
Let $X$ be a discrete random variable with $V(X)=4$, i.~e. variance equal to 4.
What is the variance of $6X+10$?

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 144}$     &
{\bf b)} & ${\Ds 24}$ &
{\bf c)} &  ${\Ds 34}$  &
{\bf d)} & ${\Ds 154}$ &
{\bf e)} & ${\Ds 400}$
\end{tabular}
\vspace{2cm}

\item
10\% of the nails produced by a certain machine have defects. If 20 nails
are randomly
selected one at a time for inspection, as the machine produces them, find
the variance of
the number of inspected nails that have defects.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 2 }$     &
{\bf b)} & ${\Ds 1.8}$ &
{\bf c)} &  ${\Ds \sqrt{1.8}}$  &
{\bf d)} & ${\Ds (1.8)^2}$ &
{\bf e)} & ${\Ds 18}$
\end{tabular}
\vspace{2.5cm}

\item
Let $Y$ denote a random variable having a geometric distribution,  with
probability of success
on any trial given by $p=0.3$.
Find $P(Y=4)$.

\begin{tabular}{cp{3cm}cp{3cm}cp{3cm}cp{3cm}}
{\bf a)} & ${\Ds (0.7)^3(0.3)}$     &
{\bf b)} & ${\Ds 1-(0.3)^3(0.7)}$ &
{\bf c)} &  ${\Ds 1-(0.7)^3(0.3)}$  &
{\bf d)} & ${\Ds 1-{3\choose 1} (0.3)(0.7)^3}$ \\[3ex]
{\bf e)} & ${\Ds 1-0.3-(0.7)(0.3)-(0.7)^2(0.3)}$
\end{tabular}
\vspace{2.5cm}

\item
Sixty percent of a population of consumers is reputed to prefer Brand A
toothpaste.
If a group of consumers is interviewed, what is the probability that
exactly 5 people must be
interviewed before a consumer is encountered who prefers Brand A?

\begin{tabular}{cp{3cm}cp{3cm}cp{3cm}cp{3cm}}
{\bf a)} & ${\Ds 4(0.4)^4(0.6)}$     &
{\bf b)} & ${\Ds {4\choose 1}(0.4)^4(0.6)}$ &
{\bf c)} &  ${\Ds 5(0.4)^4(0.6)}$  &
{\bf d)} & ${\Ds (0.6)^4(0.4)}$ \\[3ex]
{\bf e)} & ${\Ds (0.4)^4(0.6)}$
\end{tabular}
\vspace{2.5cm}

\newpage
\item
5\% of the lightbulbs manufactured on a certain assembly line are defective.
If bulbs are randomly selected one at a time  and tested, find the
probability that the third
nondefective lightbulb is found on the fifth trial.

\begin{tabular}{cp{4cm}cp{4cm}cp{4cm}}
{\bf a)} & ${\Ds {5\choose 3}(0.05)^2(0.95)^3}$     &
{\bf b)} & ${\Ds {4\choose 2}(0.05)^2(0.95)^3}$ &
{\bf c)} &  ${\Ds (0.05)^3(0.95)^2}$  \\[3ex]
{\bf d)} & ${\Ds {4\choose 2}(0.05)^3(0.95)^2}$ &
{\bf e)} & ${\Ds (0.05)^2(0.95)^3}$
\end{tabular}
\vspace{3.5cm}

\item
A certain manufacturer advertises batteries that will run for an average of
150 minutes, with
a standard deviation of 10 minutes.
Find the smallest interval, which contains at least 75\% of the performance
periods for
batteries of this type?
%Which of the following intervals contains at least 75\% of the performance
periods for batteries
%of this type?

\begin{tabular}{cp{3cm}cp{3cm}cp{3cm}cp{3cm}}
{\bf a)} & ${\Ds 50\leq Y\leq 250}$     &
{\bf b)} & ${\Ds 140\leq Y\leq 160}$ &
{\bf c)} &  ${\Ds 110\leq Y\leq 190}$  &
{\bf d)} & ${\Ds 130\leq Y \leq 170}$ \\[3ex]
{\bf e)} & ${\Ds 145\leq Y\leq 155}$
\end{tabular}
\vspace{4cm}

\item
Consider a binomial experiment for $n=20$ and $p=0.05$. Use the appended
table to find
$P(Y\geq3)$. {\bf Write  your answer on the cover sheet!}
\vspace{2.5cm}

\item
Let $Y$ be a discrete random variable having a  Poisson distribution with mean
$\lambda=1$.
Use the appended table to find $P(3\leq Y\leq 5)$.
{\bf Write  your answer on the cover sheet!}
\vspace{2.5cm}

\item
The number of telephone calls coming into a central switchboard of an
office building
has a Poisson distribution and averages 7 per minute.
Find the probability that at least one call will arrive within a given two
minute period.

\begin{tabular}{cp{3cm}cp{3cm}cp{3cm}cp{3cm}}
{\bf a)} &  ${\Ds 1-e^{-1}}$     &
{\bf b)} & ${\Ds 1-e^{-14}}$&
{\bf c)} &  ${\Ds e^{-14}+14e^{-14}}$  &
{\bf d)} & ${\Ds 2(1-e^{-7})}$ \\[3ex]
{\bf e)} & ${\Ds (1-e^{-7})^2}$
\end{tabular}
\vspace{2.5cm}

\item
The proportion of time $X$ that a certain computer system is in operation
during a week
is a random variable with probability density function
\[
   f(x)=\left\{ \begin{array}{ll} 4x^3 & 0\leq x\leq 1\\[1ex]  0 &
\mbox{\rm elsewhere}
                     \end{array}\right.
\]
Find $E(X)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds 4}$     &
{\bf b)} & ${\Ds 1}$ &
{\bf c)} &  ${\Ds \frac{4}{5}}$  &
{\bf d)} & ${\Ds \frac{2}{3}}$ &
{\bf e)} & ${\Ds 0}$
\end{tabular}
\vspace{3.5cm}

\item
For the computer in the last problem, find the probability that the
computer is in operation at
most half of the week, i.~e., $P(X\leq 0.5)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{15}{16}}$     &
{\bf b)} & ${\Ds \frac{1}{40}}$ &
{\bf c)} &  ${\Ds \frac{1}{4}}$  &
{\bf d)} & ${\Ds \frac{3}{64}}$ &
{\bf e)} & ${\Ds \frac{1}{16}}$
\end{tabular}
\vspace{3.5cm}

\newpage
\item
Let $X$ be a random variable with the uniform distribution
\[
   f(x)=\left\{ \begin{array}{ll} \frac{1}{3} & 0\leq x\leq 3\\[1ex]  0 &
\mbox{\rm elsewhere}
                     \end{array}\right.
\]
as its probability density function. Find $V(X)$.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{3}{4}}$     &
{\bf b)} & ${\Ds \frac{1}{4}}$ &
{\bf c)} &  ${\Ds \frac{3}{2}}$  &
{\bf d)} & ${\Ds \frac{9}{2}}$ &
{\bf e)} & ${\Ds 9}$
\end{tabular}
\vspace{3.5cm}

\item
Telephone calls coming into a certain switchboard follow a Poisson
distribution.
It is known that during a given 5-minute period, one call arrived at the
switchboard.
Find the probability that the call arrived within the first minute of this
period.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds \frac{4}{5}}$     &
{\bf b)} & ${\Ds 5e^{-5}}$ &
{\bf c)} &  ${\Ds \frac{1}{5}e^{-5}}$  &
{\bf d)} & ${\Ds \frac{1}{4}}$ &
{\bf e)} & ${\Ds \frac{1}{5}}$
\end{tabular}
\vspace{3.5cm}

\item
Assume that the number of fatal accidents on scheduled domestic passenger
airlines follows
a Poisson distribution with a mean of one fatal accident every 41 days.
Then the probability distribution for the interaccident times (time between
fatal accidents)
on scheduled domestic passenger flights is a(n):

\begin{tabular}{cp{5cm}cp{5cm}cp{5cm}}
{\bf a)} & Poisson distribution     &
{\bf b)} & binomial distribution&
{\bf c)} & exponential distribution  \\[2ex]
{\bf d)} & normal distribution &
{\bf e)} & uniform distribution
\end{tabular}
\vspace{1.5cm}

\newpage
\item
An engineer has observed that the gap times between vehicles passing a
certain point on a
highway have an exponential distribution with a mean of 10 seconds.
Find the probability that the next gap observed will be no longer than 1
minute.

\begin{tabular}{cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds e^{-6}}$     &
{\bf b)} & ${\Ds 1-e^{-6}}$ &
{\bf c)} &  ${\Ds 1-\frac{1}{10}e^{-6}}$  &
{\bf d)} & ${\Ds 1-e^{-60}}$ &
{\bf e)} & ${\Ds 1-e^{-1}}$
\end{tabular}
\vspace{3cm}

\item
Let the random variable $Y$ possess a probability density function
\[
   f(y)=\left\{ \begin{array}{ll} cy & 0\leq y\leq 2\\[1ex]  0 & \mbox{\rm
elsewhere}
                     \end{array}\right.
\]
Find $c$.

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

\end{problems}
\end{document}
