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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 Final Examination}
\\
{\bf May 4, 1999}
\\
{\bf 4:15 -- 6:15 pm} 
%{\bf 4:15 -- 6:15 p.m.}
%{\bf 4:15 --- 6:15 pm}
\end{center}
%\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This\rule[0cm]{0cm}{.8cm} Examination consists of 
29  multiple choice problems worth 5 points each. 
No partial credit will be given. 
You start with 5 points. 
Record your answers  by placing an {\sf X} through one letter for each 
problem.
\\
This booklet consists of 12 sheets of paper including the front cover 
and one blank page at the end.
You are provided with 2 sheets of tables. 
You may use a calculator and your prepared sheet of notes.
The notes and the calculator must not be passed to another student. 
\\[1.5ex]
\hline
\end{tabular}
\vspace{.7cm}

\begin{center} 
{\bf Answers to Multiple Choice Problems}
\\[1.3ex]
\begin{minipage}[t]{6cm}
{\bf \mbox{}\phantom{1}1.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf \mbox{}\phantom{1}2.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf \mbox{}\phantom{1}3.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf \mbox{}\phantom{1}4.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf \mbox{}\phantom{1}5.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf \mbox{}\phantom{1}6.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf \mbox{}\phantom{1}7.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\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]
{\bf 15.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) 
\end{minipage}\hspace{2.5cm} \begin{minipage}[t]{6cm}
{\bf 16.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 17.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 18.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 19.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 20.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 21.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 22.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 23.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 24.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 25.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 26.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 27.}\quad (a) \quad (b) \quad (c) \quad (d) \quad (e) \\[1ex]
{\bf 28.}\quad (a) \quad (b) \quad (c) \quad (d)  \\[1ex]
{\bf 29.}\quad (a) \quad (b) \quad (c) \quad (d)  \\[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}
\item
10 applicants are available for 4 identical jobs, 3 of them are male, and 7 are 
female.
A supervisor selects 4 of these applicants randomly.
What is the probability that at least 2 males are selected?

\begin{choices}
\item ${\Ds \frac{9}{56}}$ %${\Ds \frac{P^{10}_2+P^{10}_3}{P^{10}_4}} $
\item ${\Ds \frac{1}{3}}$ %${\Ds \frac{\choose{3}{2}\choose{7}{2}+\choose{3}{3}\choose{7}{1}}{\choose{10}{4}}} $    %correct
\item ${\Ds \frac{3}{5}}$ %${\Ds \frac{\choose{3}{3}\choose{7}{2}+\choose{3}{1}\choose{7}{3}}{\choose{10}{4}} }$  
\item ${\Ds \frac{31}{42}}$ %${\Ds \frac{\choose{4}{2}\choose{7}{2}+\choose{4}{3}\choose{7}{1}
                            % +\choose{4}{4}\choose{7}{0}}{\choose{10}{4}} }$
\item ${\Ds \frac{16}{105}}$ %${\Ds \frac{3!5!+2!4!}{P^{10}_4}} $
\end{choices}

\item
What is the probability that a random five letter ``word'' made (with repetitions allowed)
from the seven letters $\{\,$A,\,B,\,C,\,D,\,E,\,F,\,G$\,\}$ has all its letters different?
Assume that the letters are chosen randomly,  i.~e., that any letter chosen from the set 
$\{\,$A,\,B,\,C,\,D,\,E,\,F,\,G$\,\}$ is equally likely.
(Note: order counts, e.~g. ABCDD is a  different word than BACDD.)

\begin{choices}
\item ${\Ds \frac{P^7_5}{7^5}} $   %correct
\item ${\Ds \frac{5!}{7!}} $   
\item ${\Ds \frac{P^7_5}{7!}} $  
\item ${\Ds \frac{\choose{7}{5}}{7^5}} $
\item ${\Ds \frac{1}{P^7_5}} $
\end{choices}

\item
Assume $A$ and $B$ are events having probabilities $P(A)=0.5,\,P(B)=0.2$, and 
$P(AB)=0.1$.
Calculate $P(B\,|\,\bar{A})$.

\begin{choices}
\item $0.8 $
\item $0.2 $    %correct
\item $1 $  
\item $0.25 $
\item $0.4 $
\end{choices}

\newpage
\item
At a certain research institute 50\% of the members have cookies in the afternoon. 
60\% of the members have tea in the afternoon, and 15\% of the members have neither.
What fraction of the members have both cookies and tea? 

\begin{choices}
\item $95\% $    
\item $85\% $  
\item $75\% $
\item $15\% $
\item $25\% $  %correct
\end{choices}

\item
A blood typing test predicts type A for 39\% of a given population. 
41\% of the population actually has type A blood.
If a person from the population has type A blood, then the test will predict type A blood for
that person with probability 0.88.
If the test gives the type of a person's blood as A, what is the probability that the blood is 
actually type A? 

\begin{choices}
\item ${\Ds \frac{0.88\cdot0.39}{0.41}} $
\item ${\Ds 0.88\cdot0.39} $    
\item ${\Ds 0.88\cdot0.41} $  
\item ${\Ds \frac{0.41}{0.88}} $  
\item ${\Ds \frac{0.88\cdot0.41}{0.39}} $   %correct
\end{choices}
\vspace*{1cm}

\item
More Americans support a ban on handguns today than at any time
since 1982, a Newsweek poll released April 24, 1999, said. 
In the wake of a teenage killing spree in a Colorado
high school, 70\% of those polled think ``stricter gun control
would reduce violent crime in the United States''.
Assuming that this proportion is representative for the U.S. adults, what is the probability that
among the next 15 randomly selected adults at most 10 will be of this opinion? 

\begin{choices}
\item ${\Ds 0.972} $
\item ${\Ds 0.722} $    
\item ${\Ds 0.278} $  
\item ${\Ds 0.515} $  
\item ${\Ds 0.485} $   %correct
\end{choices}

\newpage
\item
Given a discrete random variable $X$ with probability distribution 
 \begin{tabular}{c|cccc}
    $X$   & $-\frac{1}{2}$      &    0     &   1   &    4       \\ \hline    
  $p(x)$ & $\frac{1}{4}$ &$\frac{1}{2}$ &$\frac{1}{8}$&$\frac{1}{8}$
\end{tabular}\,. 

Find $E(2X+1)$. 

\begin{choices}
\item $0 $
\item $2 $    %correct
\item $1$  
\item $0.5 $
\item $3 $
\end{choices}

\item
You roll a fair die twice.
Denote by $X$ the sum of the upper face numbers and by $Y$ the total number of sixes 
in both tosses. What is $P(X=8\,|\,Y=1)$? 

\begin{choices}
\item ${\Ds\frac{1}{5}} $
\item ${\Ds\frac{2}{5}} $   
\item ${\Ds\frac{1}{18}} $  %correct
\item ${\Ds\frac{1}{3}} $
\item ${\Ds\frac{1}{36}} $   
\end{choices}

\item
In a certain Starbucks caf\'e approximately 15\% of the customers order a cappuccino grande. 
Assuming that customers place their orders independently, what is the probability that on a certain 
day the 7th customer is the 4th one who orders a cappuccino grande?

\begin{choices}
\item ${\Ds \choose{7}{4}(0.15)^3(0.85)^4} $
\item ${\Ds (0.15)^4(0.85)^3} $    
\item ${\Ds \choose{7}{4}(0.15)^4(0.85)^3} $  
\item ${\Ds \choose{6}{3}(0.15)^4(0.85)^3} $  %correct
\item ${\Ds \choose{6}{3}(0.15)^3(0.85)^4} $
\end{choices}

\newpage
\item
The average number of oil tankers arriving each day at a certain port city is found to be 5.6.
The facilities at the port can handle at most 7 tankers per day.
What is the probability that on a given day tankers have to be turned away, assuming that 
arrivals follow a Poisson distribution?

\begin{choices}
\item $0.797 $
\item $0.670 $    
\item $0.330 $  
\item $0.203 $  %correct
\item $0.114 $
\end{choices}

\item
Let $X$ be a continuous random variable having probability density function
\[
       f(x)=\left\{\begin{array}{ll} \frac{3}{2}x^2 & \text{ if } -1\leq x\leq 1 \\
                                                    0 & \text{ elsewhere} \end{array}\right.
\]
Calculate $E(X)$.

\begin{choices}
\item $1 $
\item $0 $    %correct
\item ${\Ds \frac{3}{4}} $  
\item ${\Ds \frac{3}{8}} $
\item ${\Ds \frac{1}{2}}  $
\end{choices}

\item
Let $Y$ be a continuous random variable having probability density function
\[
       f(y)=\left\{\begin{array}{ll} \frac{y}{2} & \text{ if } 0\leq y\leq 2 \\
                                                    0 & \text{ elsewhere} \end{array}\right.
\]
Calculate $V(Y)$.

\begin{choices}
\item ${\Ds \frac{2}{3}} $
\item $2 $    
\item ${\Ds \frac{2}{9}} $  %correct
\item $4 $
\item ${\Ds \sqrt{\frac{2}{3}}} $
\end{choices}

\item
Let $X$ be a continuous random variable with probability density function
\[
       f(x)=\left\{\begin{array}{ll} \frac{x^2}{3} & \text{ if } -1\leq x\leq 2 \\
                                                    0 & \text{ elsewhere} \end{array}\right.
\]
Calculate $P(-1\leq X\leq 1)$. 

\begin{choices}
\item $0 $
\item ${\Ds \frac{1}{9}} $    
\item ${\Ds \frac{2}{9}} $  %correct
\item $2 $
\item ${\Ds \frac{2}{3}} $
\end{choices}

\item
The pH of water samples from a specific lake is a continuous random variable $X$ with $E(X)=5.5$ 
and $V(X)=0.16$.
Having no other information about the probability density function of $X$, which of the
following intervals should contain at least $\frac{8}{9}$ of the pH measurements? 

\begin{choices}
\item $(5.1,\,5.9) $
\item $(4.7,\,6.3) $    
\item $(5,\,6) $  
\item $(4.3,\,6.7) $  %correct
\item $(4.9,\,6.1) $
\end{choices}

\item
Suppose that a system contains a certain type of component whose time in years to failure
is given by $T$.
The random variable $T$ is modeled nicely by the exponential distribution with mean time to 
failure $\theta=5$.
If four of these components are installed in different systems, what is the probability that all four
are still functioning at the end of 5 years?

\begin{choices}
\item $0.5 $
\item ${\Ds \frac{1}{16}} $    
\item ${\Ds e^{-5}} $  
\item ${\Ds e^{-4}} $  %correct
\item ${\Ds e^{-\frac{4}{5}}} $
\end{choices}

\newpage
\item
Given the normally distributed variable $X$ with $E(X)=5$ and $V(X)=16$. 
Find $P(1\leq X\leq 2)$. 

\begin{choices}
\item $0.4321 $
\item $0.6147 $   
\item $0.0679 $  %correct
\item $0.7266 $
\item $0.9321 $   
\end{choices}
\vspace*{1cm}

\item
The loaves of rye bread distributed to local stores by a certain bakery have an average length 
of 30 centimeters and a standard deviation of 2 centimeters.
Assuming that the lengths are normally distributed, what fraction of the loaves are longer
than 31.7 centimeters? 

\begin{choices}
\item $0.1977 $  %correct
\item $0.3023 $    
\item $0.3170 $  
\item $0.4995 $
\item $0.4582 $
\end{choices}
\vspace*{1cm}

\item
A company pays its employees an average wage of \$15.90 an hour with a standard deviation of 
\$1.50.
If the wages are approximately normally distributed and paid to the nearest cent, the highest
10\% of the wages is greater than what amount? 

\begin{choices}
\item $\$18.36 $    
\item $\$17.12 $  
\item $\$17.82 $  %correct
\item $\$16.90 $
\item $\$18.97 $
\end{choices}

\newpage
\item
Let $X,\,Y$ be random variables with 
\[
    E(XY)=7,\,E(X)=2,\,E(Y)=4,\,V(X)=4,\,V(Y)=25. 
\]
Calculate the correlation coefficient $\rho$ of $X$ and $Y$.  

\begin{choices}
\item $0.1 $
\item $-0.1 $   %correct
\item $0.39 $  
\item $-0.15 $
\item $0 $   
\end{choices}

\item
Let $X$ and $Y$ be two discrete random variables having the following joint probability distribution.
\begin{center}
\begin{tabular}{p{.6cm}|p{.6cm}|p{1cm}p{1cm}p{1cm}|p{.7cm}}
             &           & \multicolumn{3}{|c|}{$X$}&    \\ \hline
             &            &  \ -1   & \  1     & \ 2      &    \\ \hline
             & \   0    &  0.02    & 0.03   & 0.05    &   \\ 
     $Y$   &  \  1    & 0.06   & 0.09    &  0.15     &     \\ 
             &  \   4    & 0.12  & 0.18    &  0.3   &    \\ \hline
            &         &         &        &         &   %\rule[-.3cm]{0cm}{1cm}
 \end{tabular}
\end{center}
Exactly one of the following statements is true. Which one is it? 

\begin{choices}
\item $P(X=1\,|\,Y=4)=0.6 $   
\item $P(Y=1)=0.4 $  
\item $X$ and $Y$ are independent  %correct
\item $P(X+Y=0)=0.02 $
\item $E(Y)=3 $   
\end{choices}

\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}

\newpage
\item
Of the customers entering a certain bike store only 20\% make purchases.
If 100 customers enter the store tomorrow, find the approximate probability that at least
30 make purchases.

\begin{choices}
\item $0.0043 $
\item $0.0064 $    
\item $0.4912 $  
\item $0.2776 $
\item $0.0088 $  %correct
\end{choices}
\vspace*{1cm}

\item
For a random sample of 25 observations, taken from a normal population 
with variance $\sigma^2=6$, find the probability $P(3.462\leq S^2\leq 10.745)$. 

\begin{choices}
\item $0.995 $
\item $0.945 $    
\item $0.895 $  
\item $0.94 $   %correct
\item $0.95 $
\end{choices}
\vspace*{1cm}

\item
A random sample of 12 graduate students of a certain secretarial school typed an average of 
79.3 words per minute with a standard deviation of 7.8 words per minute.
Assuming a normal distribution for the number of words typed per minute, find the 95\%
confidence interval for the average number of words typed by all graduates of this school.

\begin{choices}
\item $(74.34,\,84.26) $  %correct
\item $(74.39,\,84.21) $    
\item $(77.53,\,81.07) $  
\item $(75.60,\,83.00)$   
\item $(75.26,\,83.34) $
\end{choices}

\newpage
\item
In a random sample of 1000 homes in a certain city, it is found that 228 are heated by oil. 
Find the 99\% confidence interval for the true proportion of homes in this city that are heated 
by oil. 

\begin{choices}
\item $(0.202,\,0.254) $
\item $(0.194,\,0.262) $    %correct
\item $(0.206,\,0.25) $  
\item $(0.215,\,0.241) $
\item $(0.221,\,0.235) $
\end{choices}

\item
The following data, recorded in days, represent the length of time to recovery for patients 
randomly treated with one of two medications to clear up severe bladder infections:
\\[1ex]
\begin{tabular}{l|lll}
   Medication 1 & $n_1=16$ & $\bar{x}_1=19$ & $s_1^2=1.8$ \rule[-.2cm]{0cm}{.6cm}\\ \hline
   Medication 2 & $n_2=14$ & $\bar{x}_2=17$ & $s_2^2=1.5$ \rule[-.2cm]{0cm}{.6cm}
\end{tabular}

Find the 99\% confidence interval for the difference $\mu_1-\mu_2$ in the mean recovery time 
for the two medications, assuming normal populations with equal variances. 

\begin{choices}
\item $(0.697,\,3.303) $  %correct
\item $(0.836,\,3.164) $    
\item $(0.785,\,3.215) $  
\item $(0.651,\,3.349) $
\item $(1.88,\,2.12) $  
\end{choices}

\item
A marketing expert for a pasta-making company believes that at most 30\% of pasta lovers 
prefer  lasagna.
It is found that 37 out of 100 pasta lovers choose lasagna over other pastas.
Which of the following is the smallest significance level with which we can reject  the null hypothesis 
that at most 30\% of pasta lovers prefer lasagna?

\begin{choices}
\item $0.437 $
\item $1.528 $    
\item $0.152$  
\item $0.126 $
\item $0.063 $   %correct
\end{choices}

\newpage
\item
A random sample of 50 bags of White Cheddar Popcorn weighed, on average, 5.39 ounces with a 
standard deviation of 0.51 ounces. 
Is there enough evidence to conclude that the average weight of a popcorn bag is less than 5.5 
ounces?
Test the null hypothesis $H_0:\,\mu\geq 5.5$ at the significance levels $\alpha=0.05$ and
$\alpha=0.1$.  

\begin{choices}
\item Reject $H_0$ at $\alpha=0.1$ and don't reject $H_0$ at $\alpha=0.05$ %correct
\item Reject $H_0$ at $\alpha=0.05$ and don't reject $H_0$ at $\alpha=0.1$
\item Reject $H_0$ at $\alpha=0.1$ and at $\alpha=0.05$ 
\item Don't reject $H_0$  at $\alpha=0.1$ and don't reject $H_0$  at $\alpha=0.05$
\end{choices}

\item
In the American Heart Association journal {\em Hypertension}, researchers report that
individuals who practice Transcendental Meditation (TM) lower their blood pressure 
significantly. 
A random sample of 20 male TM practitioners meditate on average for 8.6 hours per week 
with a standard deviation of 2.02 hours. 
Test the null hypothesis that, on average, men who use TM meditate for at most 8 hours at
the significance levels $\alpha=0.05$ and $\alpha=0.01$. 
Assume normal distribution for the meditation time per week. 

\begin{choices}
\item Reject $H_0$ at $\alpha=0.05$ and don't reject $H_0$ at $\alpha=0.01$ %correct
\item Reject $H_0$ at $\alpha=0.01$ and don't reject $H_0$ at $\alpha=0.05$
\item Reject $H_0$ at $\alpha=0.01$ and at $\alpha=0.05$ 
\item Don't reject $H_0$  at $\alpha=0.01$ and don't reject $H_0$  at $\alpha=0.05$
\end{choices}








\end{problems}
\end{document}
