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\hskip 1.5in \textbf{Final Exam--Stat 100--Spring 1997}   
\vskip .2in  
You may use calculators, and ONE page of notes (writing allowed  
both sides). Each problem is 
worth 20 points. Different parts of a problem have equal weight unless 
otherwise indicated.  Do not spend too much time on any one problem. 
 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. 

\renewcommand{\labelenumi}{\alph{enumi}.}
\vspace{.3in}


1. Consider the data -3, 2, 2, -4, -1, 1  . 

\hspace{.2in}  a. (14 pts.)  
Determine the sample mean, median and  sample standard deviation.

\hspace{.2in}  b. (6 pts.) 
 How do the sample mean, median and  sample standard 
deviation change
if we add 5 to every number in the above data?

\vspace{.3in}
2. Lisa wonders about typical July weather in Springfield.
Studying old July weather reports, Lisa classifies days 
in three ways: {\bf \ sunny} vs. {\bf cloudy;\ } {\bf hot} vs. 
{\bf cold;\ }  
{\bf humid} vs. {\bf dry.}
Lisa finds that in July 

\hspace{.2in} ---
it is {\bf sunny} with probability 0.6  

\hspace{.2in} ---
{\bf cloudy} days are {\bf humid} with probability 0.7

\hspace{.2in} ---
it is {\bf cold} with probability 0.4

\hspace{.2in} ---
it is {\bf hot} and {\bf humid} with probability 0.2

\hspace{.2in} ---
it is {\bf cold} and {\bf humid} with probability 0.3

\hspace{.2in} ---
{\bf sunny} skies and {\bf hot} weather are independent. 

\vspace{.1in}
What does Lisa conclude for

\hspace{.2in} a. 
the probability that it is {\bf cloudy}

\hspace{.2in} b. 
the probability that it is {\bf cloudy} and {\bf humid}

\hspace{.2in} c. 
the probability that a {\bf dry} day is {\bf humid}

\hspace{.2in} d. 
the probability that it is {\bf sunny} or {\bf hot} or {\bf both}

\hspace{.2in} e. 
the probability that on a {\bf hot} day it is {\bf humid} ? 




\vspace{.3in}
3. Suppose the number $X$ of raisins in a muffin from Jim's Bakery
has a probability distribution
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline 
$n$ & 0 & 1 & 2 & 3 & 4 & 5 or more    \\ \hline
$P[X=n]$ & 0.1 & 0.2 & 0.3 & 0.3 & 0.1 & 0 
 \\ \hline
\end{tabular}
\end{center}
\vspace{.1in} 
\hspace{.2in} 
a. What is the probability that a muffin has fewer than 3 raisins?

\hspace{.2in} 
b. Find the mean of $X$.

\hspace{.2in} 
c. Find the standard deviation of $X$. 

\hspace{.2in} 
d. Buying 250 muffins from Jim's, 
how many raisins does one expect to get in total?


\vspace{.3in}
 
%4. In a box there are 4 glazed and 7 powdered doughnuts. Homer 
%selects 6 doughnuts at random.
%\begin{enumerate}
%\item
%Find the probability that Homer selects exactly 2 glazed doughnuts.
%\item
%Find the probability that no more than 1 of the selected doughnuts
%is glazed.
%\end{enumerate}

4.
An advertisement seeking volunteers for a clinical research has
4 respondents below age 40 and 7 above age 40. The researcher
selects 5 persons at random for testing a new treatment.

\vspace{.1in} 
\hspace{.2in} a. (7 pts.) 
How many different selections are possible?

\hspace{.2in} b. (8 pts.) 
What is the probability that exactly 3 of the selected persons are below 40?

\hspace{.2in} c. (5 pts.) 
What is the probability that 3 or more of the selected persons are below 40?

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

5. Suppose $40\%$ of the students at the University have a part-time job.

a. (8 pts.)  In a random sample of 15 students, 
what is the probability that 
at least 4 have a part-time job?

b.  For a random sample of 150 students, let $X$ denote the number 
with a part-time job.  

\hspace{.2in} \ i.(3 pts.)  What is the exact distribution of $X$?

\hspace{.2in} ii.(9 pts.)  Approximate $P(X<72)$.


\vspace{.25in}

6. Let $\overline{X}$ denote the sample mean 
for a random sample of size 50 from a normal population with 
 mean 10 and standard deviation 2.  

%6. Suppose 50 samples are taken from the normal distribution with mean 10
%and standard deviation 2.  Let $\overline{X}$ denote the sample mean.

a. (8 pts.)  What is the distribution of $\overline{X}$?  Is this 
distribution  exact or approximate? 

b. (6 pts.)  Find $P(\overline{X} > 10.1)$.

c. (6 pts.)  Find the number $b$ so that $P(\overline{X}<b)=.7939$.
%c.(7 pts.)  Find the number $b$ so that $P(\overline{X}<b)=.67$.

\vspace{.25in}

7.
A factory produces new VCR's.  In a set of 600 VCR's produced by the
factory, 24 are found to be defective.  

a. (16 pts.)  Determine a $95\%$ confidence interval for the proportion of
defective VCR's produced by the factory.

b. (4 pts.)  Suppose the factory claims that the probability
of a VCR it produces being defective is .015.  Would you believe the
factory based on the confidence interval found in part a?  Explain
your response.   


\vspace{.25in}

8. The SunBrite Battery Company claims its new type of battery  
lasts longer. The old type has an average usage life of 8 hours. 
In a sample of  15 new-type batteries,  the average usage life is 
8.5 hours with standard deviation 0.75 hours. 
 

 
%Suppose a battery company decides to try to improve its product.
%Fifteen improved batteries are tested and the average usage time is
%8.5 hours and the standard deviation is 0.75 hours.  The old type of
%batteries had an average usage time of 8 hours.  

 a. (4 pts.)  State the Null and Alternative Hypotheses.

 b. (6 pts.)  State the test statistic and rejection region for a test 
with a $5\%$ significance level.

 c. (5 pts.)  Determine if the claim of longer life is substantiated 
at the $5\%$
significance level.

 d. (5 pts.)  Which of the following best approximates 
 the P-value:
$\ \ .10,\  .05,\  .025,\  .01,\  .005
\ ?$


\vspace{.25in}

9. From a certain obese population,  
ten persons are randomly selected  for weight reduction program A, 
and ten  are randomly selected  for program B.  After one year, 
the sample statistics for weight loss in pounds for the two 
programs are  

\vspace{.1in}
\hspace{1.9in} A \hspace{.5in} B 

\hspace{.6in} mean   \hspace{.75in} 25 \hspace{.5in} 30 

\hspace{.6in} stand. dev.    \hspace{.3in} 25 \hspace{.5in} 15 

\vspace{.1in}
 a. (15 pts.) Construct a 90\% confidence interval for $\mu_A-\mu_B$, 
the expected 
difference in weight loss under the two programs.

 b. (5 pts.) State any assumption or computation you make to justify 
your procedure.

\vspace{.2in} 
10. A poll of 350 voters on a proposed law gives the following data. 
\begin{center}
\begin{tabular}{|c||c|c|c|c|} \hline 
$\qquad $ & Favor  & Indifferent  & Oppose  & Total    \\ \hline
Male & 93 & 72 & 21 & 186 \\ \hline 
Female & 55 & 79 & 30 & 164  \\ \hline 
Total & 148 & 151 & 51 & 350 
 \\ \hline
\end{tabular}
\end{center}
Test the claim that the proposed law appeals differently to 
men and women, at $\alpha = .05$. 
 
%9. Two fitness clubs offer weight training programs for men.  

%A certain fitness club claims that its  
%Pain-and-Gain weight training program is superior to the  
%competitors' Hardbody weight training  program.  Five pairs of men 
%of comparable strength and size are selected, and each pair is randomly 
 
%is randomly assigned to each of the programs. After six months, the 
%\vspace{.3in}
%10. To compare the effectiveness of four drugs in relieving postoperative 
%pain, 200 patients are randomly split into four groups of 50, and each 
%group receives one of the drugs. The number of patients pain-free five 
%hours after surgery is 25 for Drug 1, 30 for Drug 2, 22 for Drug 3, 
%and 26 for Drug 4. 

%Test the null hypothesis that all four drugs are equally effective, 
%at significance level $\alpha = .05$.  
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