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\hskip 1.5in \textbf{Final Exam--Stat 100--Spring 2001} 
\vskip .2in  
You may use calculators, and ONE page of notes (writing allowed  
both sides). Each problem is 
worth 20 points.  
 Put a box around the final answer to a question. 
EACH PROBLEM SHOULD BE ANSWERED ON A SEPARATE PAGE. 
Use the back side of 
a page if necessary. 
Don't spend too much time on any one problem. 
\renewcommand{\labelenumi}{\alph{enumi}.}
\vspace{.2in}


1.
Let $A$ and $B$ be two events. The probability that $A$ occurs is $.2$, the
probability that $B$ does not occur is $.7$, and the probability that $A$ or
$B$ occurs is $.4$. 

\hspace{.2in}  a. 
What is the probability that $B$ occurs?

\hspace{.2in} b. 
What is the probability that both $A$ and $B$ occur? 

\hspace{.2in} c. 
Are the events $A$ and $B$ independent?
Briefly justify your answer. 

\hspace{.2in} d. What is the probability that neither 
$A$ nor $B$ occurs? 

\vspace{.3in}

2. 
From a group of 12 people, 4 are chosen randomly to 
work on Project A, and 4 are chosen randomly 
and independently to work on Project B. 

What is the probability that none of the 12 works 
on both projects? 

\vspace{.2in} 

3. 
Suppose the number of speeding tickets $X$ issued in a workday 
by Officer Joe 
 has the probability distribution 

\begin{center}
\begin{tabular}{|c||c|c|c|c|} \hline 
$x$ & 0 & 1 & 2 & 3     \\ \hline
$\textnormal{Prob}[X=n]$ & 0.2 & 0.3 & 0.4 & 0.1 
 \\ \hline
\end{tabular}
\end{center}

\hspace{.2in} a. 
  What is $E(X)$? 

\hspace{.2in} b. 
What is the standard deviation  of $X$? 

\hspace{.2in} c. 
What is the expected number of speeding tickets Officer Joe will issue 
in his next 100 days of work? 

\hspace{.2in} d. 
 Suppose the numbers of tickets issued on 
different days is independent. What is the probability that 
Officer Joe will issue exactly one ticket over the next four days? 
\vspace{.2in} 

4. 
Suppose that the probability a voter will believe a rumor about 
a politician is 1/5. 

\hspace{.2in} a. (5 points) If 20 voters are told the rumor, what is the 
probability at least 3 believe it? 

\hspace{.2in} b. (5 points)  If 20 voters are told the rumor, what 
is the expected number of believers? 

\hspace{.2in} c. (10 points) If 400 voters are told the rumor, estimate  the 
probability that fewer than 70 believe it. 
 


\vspace{.2in}


5. 
The amount of dirt picked up in one scoop by a diesel shovel has 
expected value 25 pounds with standard deviation 4 pounds. 

\hspace{.2in} 
In 100 scoops, approximately what is the probability that 
the shovel will pick up less than 2520 pounds of dirt? 




\vspace{.25in}

6. 
A scientist is 
studying the effect of communication modes on the problem solving
capability of teams.  
For one mode of communication a random sample of 35 teams
completed a specific 
task in an average of 25.7 minutes with a standard deviation
of 3.8 minutes.  

\hspace{.2in} 
Construct a 90\% confidence interval for the mean time in minutes 
to complete the task. (Substitute correctly into the appropriate formula, 
but do not spend time doing the final calculator computations.) 



\vspace{.25in}

\hspace{.2in} ************THERE ARE MORE PROBLEMS ON THE BACK SIDE************
\pagebreak

7. 
An educational psychologist is studying differences in intelligence scores
produced by two test procedures, A and B.  Four subjects 
were randomly selected and 
given both tests.  The resulting scores are in the following table 
(each column gives the scores of one subject). 



\begin{center}
\begin{tabular}{|c||c|c|c|c|} \hline 
Test A & 98 & 84 & 91 & 83       \\ \hline
Test B & 86 & 86 & 82 & 84   
 \\ \hline
\end{tabular}
\end{center}



Assume the population of test score differences is approximately normal. 
Construct an appropriate
 test to determine whether the mean scores are equal,  at
level of significance $\alpha =.10$  

\hspace{.2in} 
(a) Formulate the null and alternative hypothesis

\hspace{.2in} 
(b)  Write the test statistic

\hspace{.2in} 
(c)  State the rejection region

\hspace{.2in} 
(d)  Compute the test statistic for the sample data and estimate 
the P-value. 

\hspace{.2in} (e)  State your conclusion.  
Do you  conclude that the mean scores for the two
tests differ?

 
\vspace{.25in}

8. 
Ten rattle snakes in Arizona are found to have a mean length of $1.2$ meters
with a standard deviation of $0.25$. In California a sample of 8 rattlesnakes
is found to have a mean length of $0.9$ meters with a standard deviation of 
$0.6$ meters. Construct a 98\% confidence interval for the mean difference in
lengths,  $\mu_A - \mu_C$. 
 (Substitute correctly into the appropriate formula, 
but do not spend time doing the final calculator computations.) 

\hspace{.2in} What assumptions do you make to justify your procedure?  


\vspace{.25in}




9. 
 In a study on the effect of diet and life-style on heart
disease, 96 patients with severe coronary blockage were randomly
assigned, 48 to an experimental group and 48 to a control group.
The patients in the experimental group had a low-fat vegetarian
diet, regular exercise, and stress-management training, whereas
those in the control group had a low-fat diet and moderate
exercise. The condition of their coronary blockage was monitored
throughout the study period, and the following results were noted.

\begin{center}
\begin{tabular}{|c||c|c|c|c|c||c|} \hline 
 & Worse & No Change & Improved & Total \\ \hline 
Experimental & 4& 8 & 36 & 48      \\ \hline
Control      & 8& 25& 15 & 48 \\ \hline 
 Total       & 12& 33& 51& 96 \\
 \hline
\end{tabular}
\end{center}
\vspace{.1in} 
 
a. (18 pts.) 
Analyze the data to determine if the changes in coronary blockage
were significantly different for the two groups of patients.
Test at significance level $\alpha = .05$. 
State your conclusion clearly. 
Is the evidence very strong? 

b. (2 pts.) 
What rule-of-thumb criterion justifies the use of the test you 
used? 

 \vspace{.25in} 

10. Answer each part TRUE or FALSE. No justification required. 

\hspace{.2in} 
a. If  $X_1$ and $X_2$ are normally distributed random variables 
with respective means $\mu_1,\mu_2$ and standard deviations 
$\sigma_1 ,\sigma_2$, then 
 $\textnormal{Prob}(|X_1-\mu_1|< 3\sigma_1)=\textnormal{Prob}
(|X_2 - \mu_2| < 3\sigma_2)$. 
 
\hspace{.2in} 
b. The significance level of a hypothesis test is the probability 
of incorrectly choosing the alternate hypothesis. 

\hspace{.2in} 
c. If $X$ is a random variable with the Binomial($n,p$) distribution, 
with $n=1000$ and $p=.999$, then it is appropriate to use the 
normal approximation for this distribution. 

\hspace{.2in} 
d. If we run a hypothesis test  at level of significance 
$\alpha = .1$,  and the $P$-value is .2 , then we should reject the 
null hypothesis. 

\hspace{.2in} 
e. If $\overline X_n$ denotes the average salary of a random sample 
of $n$ persons from a certain neighborhood, then 
$E(\overline X_{900}) = (1/3)E(\overline X_{100})$. 


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