%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt]{article}
\usepackage{amssymb}
\topmargin-2.5cm
\textheight 24cm
\textwidth18.5cm
\oddsidemargin-.7cm
\evensidemargin-.7cm
%\pagestyle{empty}
\parindent0cm
\parskip 2.5ex
\newcommand{\re}{\rm Re}
\newcommand{\im}{\rm Im}
\newcommand{\C}{\Bbb C}
\newcommand{\Ds}{\displaystyle}
\newcommand{\wegdamit}[1]{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}}
\newenvironment{subproblem}{
     \begin{list}{\bf{(\alph{subprob})}\hfill}{\usecounter{subprob}
     \topsep-1.4ex \labelwidth.6cm \leftmargin.8cm \labelsep.2cm 
     \rightmargin0cm \parsep2ex \itemsep2ex
     \partopsep1.4ex}}{\end{list}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\mbox{}\vspace*{-1.5cm}

\hfill{\bf Instructor:} Heide Gluesing-Luerssen

\begin{center}
{\bf Mathematics 214: Introduction to Statistics }
\\
{\bf Review for Exam 2, March 1999}
\end{center}

\begin{problems}
\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!}

\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!}

\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}

\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}

\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}

\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}

\newpage
\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?  

\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}

\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}

\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}

\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}

\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}

\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}

\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}

\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}

\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}

\item 
Find $P(Z\leq 2)$ where $Z$ is a standard normal random variable.  
{\bf Write your answer on the cover sheet!}

\item 
Assume $X$ is a normal random variable with mean 3 and standard deviation 2. Find $P(-1\leq X\leq 5)$.
{\bf Write your answer on the cover sheet!}

\item 
The grade-point averages of a large population of college students are approximately normally distributed 
with mean equal to 2.4  and standard deviation equal to 0.5. 
Suppose that three students are randomly selected from the student body. 
What is the probability that all three will possess a grade-point average in excess of 3.0? 

\begin{tabular}{cp{2.3cm}cp{2.3cm}cp{2.3cm}cp{2.5cm}cp{2.5cm}}
{\bf a)} & ${\Ds (0.3849)^3}$     &
{\bf b)} & ${\Ds (0.1151)^3}$ &
{\bf c)} &  ${\Ds (0.8849)^3}$  & 
{\bf d)} & ${\Ds 1-(0.1151)^3}$ & 
{\bf e)} & ${\Ds 1-(0.8849)^3}$
\end{tabular}
\end{problems}

\newpage
\begin{center} 
\begin{minipage}[t]{6cm}
{\bf 1.}\quad 0.075 \\[1ex]
{\bf 2.}\quad 0.079 \\[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) \\[1ex]
{\bf 9.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) 
\end{minipage}\hspace{2.5cm} \begin{minipage}[t]{6cm}
{\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 0.9772 \\[1ex]
{\bf 17.}\quad 0.8185 \\[1ex]
{\bf 18.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
\end{minipage}
\end{center}

\end{document}