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FINAL EXAM--Stat 100--Fall 1996    
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You may use calculators and ONE page of notes. Each problem is 
worth 20 points. Do not spend too much time on any one problem. 
 Put a box around the final answer to a question. 
THERE ARE MORE QUESTIONS ON THE BACK. 
\newline \newline  
1.  Six parents had the following numbers of children: 
 	1, 2, 2, 3, 7, 3 . 
\newline  
(a) (8 pts.) Determine the sample mean and median number of children. 
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(b) (6 pts.) Determine the sample standard deviation.   
\newline 
(c) (6 pts.)  Graph a histogram for this data. For the bins (bases of 
    rectangles), use intervals [.5, 1.5), [1.5, 2.5), [2.5, 3.5), etc. 
    Indicate the scale on the y-axis. 
 \newline 
\newline 
2. A sample space is the union of events $A,B,C$.  Suppose 
\+ &\hskip .5in &     
 the events  $B$ and $C$ are mutually exclusive; \qquad \qquad 
&P($B\overline C) = .2$\ ; \cr    
\+ & &$A$ and $B$ are independent and P$(AB) = .12$\ ; &P$(C| A) = 0$\ . \cr   
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\noindent Now determine the probabilities below. 

(a) P($B$) \quad (b) P($A$)  \quad (c) P($A\cup B$) \quad 
(d) P($AC$) \quad (e) P($C$) 
%(answer: .2, .6, .68 , 0, .32)
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3. Suppose  a discrete random variable $X$ has 
the probability distribution: 
 
f(0) = .5, \ f(1) = .2, \ f(2) = .3 . 
\qquad (Here, f($x$) denotes Prob$(X=x)$.) \newline 
Compute the mean and standard deviation of X. 
\newline 
\newline 
%Binomial dist. 
4. (a)  In her morning commute, a professor passes through an 
intersection at which the traffic light is red with probability .3 .  
What is the probability she will hit 10 or more red lights 
in 18 working days? 

(b) Albert flips a fair coin 100 times. 
Approximately what is the probability that the number of heads 
he gets is 40 or less? Justify your answer. 
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\newline 
%CLT Ch. 7
5.    The distribution of 
personal  income among  workers in a certain industry 
has mean \$52,000 and standard deviation  \$4,000. 
Approximately what is the probability that in  a random sample 
of 100 workers the average income would exceed \$53,000? 
\newline \newline 
%(Large sample hypothesis test)(mean )  
6. 	A teacher claims that students at his University  
would have average score at least 75 on a certain national exam. 
A random sample of 100 students at his University score with 
average 78 and sample standard deviation 14. Is this strong evidence 
for the teacher's claim? 

	Formulate this problem as a hypothesis test with level of 
significance $\alpha = .025$.  State null and alternate hypotheses, 
test statistic and rejection region. Give an assumption or 
explanation which justifies your procedure.  
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%(Difference of 2 treatements, small sample, pooled) 
 7. Measurements of aerobic capacities are recorded for a group of 
20 Peruvian native highlanders and a group of 10 U.S. lowlanders 
acclimatized as adults in high altitudes. The following summary 
statistics were obtained from the data. 
\newline 
%(copy table p.422) 
\+ \hskip.5in\quad & \hskip1.5in &\ \ Peruvian \hskip.5in    &\qquad U.S.  \cr
\+                 &           &Highlanders  &Lowlanders             \cr
\+ \hskip.5in\quad &Mean       & \quad \  46.3 &\qquad   38.5    \cr 
\+ \hskip.5in\quad &Standard deviation 	& \quad \ \ 5.0 &\qquad   \ 5.8		\cr  
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\noindent 
(a) Construct a 95\% confidence interval for the mean difference 
    in aerobic capacity between the two groups. 
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(b) Give appropriate assumptions which justify your procedure. 
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    Does the reasonableness of any assumption depend on the summary  
    statistics? 
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8. (a)  Suppose the null hypothesis 
is rejected at  significance level $\alpha  =.05$ . 
\newline 
What, if anything, can you deduce about the possibilites
for the P-value? 
\newline 
Very briefly explain. 
 
(b) You are allowed to roll a fair die 10 times, and then you 
are given X dollars, where X is the total of your 10 throws. 
What is the expected value of X? 
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\newline 
%Matched Pair 
9. A new pumpkin fertilizer is tested. The new and old fertilizers 
are used  on pumpkins in four soil types A,B,C,D. 
The weight in pounds of the pumpkins 
after eight weeks of growth is recorded below. 
 \+ \hskip .5in \quad & Soil type \hskip1in  
&A\qquad &B\qquad &C\qquad&D\qquad	\cr
\+ \hskip .5in \quad &New fertilizer    
&3\qquad &0\qquad &7\qquad&8\qquad	\cr 
\+ \hskip .5in \quad &Old fertilizer    
&2\qquad &0\qquad &5\qquad&7\qquad	\cr  
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For a suitable test, is there evidence at the .05 level of significance that 
the new fertilizer is better than the old one? 
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\newline 
10. An M\&M producer claims  that the 
colors of M\&M's occur in equal proportions. 
A Stat 100 student wants strong evidence 
to dispute the claim. The student 
randomly samples 1,000 M\&M's from different stores, 
and obtains the counts below of the colors red (R), orange (O), 
yellow (Y), green (G) and blue (B). 
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\+ \hskip .5in &Color 
\qquad &\ R\qquad &\ O\qquad &\ Y\qquad &\ G\qquad &\ B\qquad \qquad &TOTAL \cr 
\+  &Count  &210 &230 &190 &180 &190 &1000\cr 
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Determine with  a suitable test whether the producer's claim 
is rejected at level of significance .05.  

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