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\centerline{MONT 107N -- Understanding Randomness}
\centerline{Midterm Examination -- March 19, 2010}
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{\it Directions} 
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Do all work in the blue exam booklet.  {\it Show 
all work necessary to justify your answers.}  Please turn over for question V.  
There are 100 possible points.
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I. A gambler is playing roulette. 
On each spin of the wheel, 
he places a bet $\$1$ on {\it six numbers}.
If one of his block of six numbers comes up, 
he will win $\$5$ on a $\$1$ bet. Otherwise, he loses $\$1$.
\item{A)}  (10) Construct a box model for this bet and compute the average and SD of the box.  (Recall: There are a total of 38 slots 
on the roulette wheel.)
\item{B)} (10) If this bet is played 36 times, compute the EV and the SE for the total outcome
(i.e. how much ahead or behind the gambler will be after the 36 plays).
\item{C)} (5) What are the gambler’s chances of ending up $\$5$ or more ahead after
these 36 spins of the wheel?
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II.  Assume 100 draws with replacement are made from the following box.
$$ 1 \quad 2 \quad 3 \quad 3 \quad 7 \quad 8$$                   
\item{A)} (5)  What box model would you use to model the number of 3s that are drawn?
\item{B)} (5)  If 40 3's are drawn among the 100 numbers, what is the chance
error?
\item{C)} (10)  Would you consider the chance error from part B to be a large chance error, roughly
       an average chance error, or a small chance error?   Explain your answer. 
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III.  On November 6, 2007, the Gallup polling organization
reported the results of a survey about the role of
women in society in the three North African countries of Algeria, Morocco, and Tunisia.
Among the results of the poll, $76\%$ of Tunisian adults agreed that
 “Women should be allowed to hold any job for which they are
qualified outside the home.” The Tunisian sample consisted of 912 adults.
\item{A)}  (15)   Based on this
information, construct a $95\%$ confidence interval for this estimate. (Assume for the purposes
of this calculation that it was a simple random sample.)
\item{B)}  (5)  Your answer in A should have used an estimate
for the standard error for the precentage.  What is the largest possible 
actual value for this standard error?
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IV.  (15)  An urn contains 1000 blue marbles and 500 red marbles.
The process of drawing 100 marbles with replacement from the urn and 
recording the number of red marbles is repeated 10 times.  The counts are
$$28, 29, 27, 31, 34, 29, 26, 32, 27, 30$$
Is there any reason to be suspicious about the results?  Answer yes
or no and back up your claim with calculations. 
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V.  Short answer.
\item{A)}  (5)  For a final project in a psychology class, 
two students want look at the connection between a person's long-term memory after
hearing lists of words read aloud to them and the ``emotional content'' of those words.
Student A says, ``it is fine if we use the students in our 
class as the subjects for the study.''  Student B says, ``it would be 
better to put an ad in the college newspaper to recruit the subjects.''
Are there possible problems with both of these methods?  If not, which would be better?
If so, can you suggest a clearly superior way of choosing the subjects?  
\item{B)}  (5)  Why are we justified in using the normal curve to analyze the chances of 
sums of draws from a box even when the box itself does not have a normal distribution
of values?  Is there a limitation to keep in mind when doing this?
\item{C)}  (5)  True or False and explain:  All estimated $95\%$ confidence intervals
for a population percentage computed from samples of the same size $n = 100$
from the same population will have the same width.
\item{D)}  (5) True or False and explain: 
A simple random sample of size 300 is selected from  
the 27000 students at a large state university and a
$95\%$ confidence interval for the percentage is estimated
as $74\% \pm 5\%$.  These results show that it is impossible for
$80\%$ of the student population to be undergraduates.   
\bye
