# Once upon a time, I had a fast-food lunch with a mathematician colleague

1.
(10 points) Once
upon a time, I had a fast-food lunch with a mathematician colleague. I noticed
a very strange behavior in him. I called it the Au-Burger Syndrome since it was
discovered by me at a burger joint. Based on my unscientific survey, it is a rare
but real malady inflicting 2% of mathematicians worldwide. Yours truly has
recently discovered a screening test for this rare malady, and the finding has
just been reported to the International Association of Insane Scientists (IAIS)
for publication. Unfortunately, my esteemed colleagues who reviewed my
submitted draft discovered that the reliability of this screening test is only
80%. What it means is that it gives a positive result, false positive, in 20%
of the mathematicians tested even though they are not afflicted by this

I
have found an unsuspecting victim, oops, I mean subject, down the street. This
good old mathematician is tested positive! What is the probability that he is
actually inflicted by this rare disabling malady?

2.
(5 points) Most
of us love Luzon mangoes, but hate buying those that are picked too early.
Unfortunately, by waiting until the mangos are almost ripe to pick carries a
risk of having 15% of the picked rot upon arrival at the packing facility. If
the packing process is all done by machines without human inspection to pick
out any rotten mangos, what would be the probability of having at most 2 rotten
mangos packed in a box of 12?

3.  (5 points)
We have 7 boys and 3 girls in our church choir. There is an upcoming concert in
the local town hall. Unfortunately, we can only have 5 youths in this
performance. This performance team of 5 has to be picked randomly from the crew
of 7 boys and 3 girls.

a. What is the probability that all 3 girls are picked
in this team of 5?

b. What is the probability that none of the girls are
picked in this team of 5?

c. What is the probability that 2 of the girls are
picked in this team of 5?

4.
(10 points) In
this economically challenging time, yours truly, CEO of the Outrageous Products
Enterprise, would like to make extra money to support his frequent filet-mignon-and-double-lobster-tail
dinner habit. A promising enterprise is to mass-produce tourmaline wedding
rings for brides. Based on my diligent research, I have found out that women's
ring size normally distributed with a mean of 6.0, and a standard deviation of
1.0. I am going to order 5000 tourmaline wedding rings from my reliable Siberian
source. They will manufacture ring size from 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0,
7.5, 8.0, 8.5, 9.0, and 9.5. How many wedding rings should I order for each of
the ring size should I order 5000 rings altogether? (Note: It is natural to
assume that if your ring size falls between two of the above standard
manufacturing size, you will take the bigger of the two.
)

5.
(5 points) A soda company want to stimulate sales in this economic climate by
giving customers a chance to win a small prize for ever bottle of soda they
buy. There is a 20% chance that a customer will find a picture of a dancing
banana () at the bottom of the cap upon opening up a bottle of soda. The
customer can then redeem that bottle cap with this picture for a small prize.
Now, if I buy a 6-pack of soda, what is the probability that I will win
something, i.e., at least winning a single small prize?

6. (5 points) When constructing
a confidence interval for a population with a simple random sample selected
from a normally distributed population with unknown σ, the Student t-distribution
should be used. If the standard normal distribution is correctly used instead,
how would the confidence interval be affected?

7. (10 points) Below is a
summary of the Quiz 1 for two sections of STAT 225 last spring. The questions
and possible maximum scores are different in these two sections. We notice that
Student A4 in Section A and Student B2 in Section B have the same
numerical score.

8. (5 points) My brother wants
to estimate the proportion of Canadians who own their house. What sample size
should be obtained if he wants the estimate to be within 0.02 with 90%
confidence if

a. he uses an estimate of 0.675
from the Canadian Census Bureau?

b. he does not use any prior
estimates? But in solving this problem, you are actually using a form of
"prior" estimate in the formula used. In this case, what is your
"actual" prior estimate? Please explain.

9. (5 points) An amusement park
is considering the construction of an artificial cave to attract visitors. The
proposed cave can only accommodate 36 visitors at one time. In order to give
everyone a realistic feeling of the cave experience, the entire length of the
cave would be chosen such that guests can barely stand upright for 98% of the
all the visitors.

The mean height of American men
is 70 inches with a standard deviation of 2.5 inches. An amusement park
consultant proposed a height of the cave based on the 36-guest-at-a-time
capacity. Construction will commence very soon.

The park CEO has a second
thought at the last minute, and asks yours truly if the proposed height is
appropriate. What would be the proposed height of the amusement park
consultant? And do you think that it is a good recommendation? If not, what
should be the appropriate height? Why?

10. (5 points) A department
store manager has decided that dress code is necessary for team coherence. Team
members are required to wear either blue shirts or red shirts. There are 9 men
and 7 women in the team. On a particular day, 5 men wore blue shirts and 4
other wore red shirts, whereas 4 women wore blue shirts and 3 others wore red
shirt. Apply the Addition Rule to determine the probability of finding
men or blue shirts in the team.

Please refer to the
following information for Question 11 and 12.

It is an open secret that
airlines overbook flights, but we have just learned that bookstores underbook
(I might have invented this new term.) textbooks in the good old days that
we had to purchase textbooks.

To make a long story short,
once upon a time, our UMUC designated virtual bookstore, MBS Direct, routinely,
as a matter of business practice, orders less textbooks than the amount
requested by UMUC's Registrar's Office. That is what I have figured out.......
Simply put, MBS Direct has to "eat" the books if they are not sold.
Do you want to eat the books? You may want to cook the books before you eat
them! Oops, I hope there is no account major in this class?

OK, let us cut to the
chase..... MBS Direct believes that only 85% of our registered students will
stay registered in a class long enough to purchase the required textbook. Let's
pick on our STAT 200 students. According to the Registrar's Office, we have 600
students enrolled in STAT 200 this spring 2014.

11. (10 points) Suppose you are
the CEO of MBS Direct, and you want to perform a probability analysis. What
would be the number of STAT 200 textbook bundles you would order so that you
stay below 5% probability of having to back-order from Pearson Custom
Publishing? (Note: Our Provost would be very angry when she hears that
textbook bundles have to be back-ordered.
In any case, we no longer need
the service of MBS Direct as we are moving to 100% to free eResources. Auf
Wiedersehen
, MBS Direct......)

IMPORTANT: Yes, you may use
technology for tacking Question 11

12. (5 points) Is there an
approximation method for Question 11? If so, please tackling Question 11 with
the approximation method.