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

(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 horribly-embarrassing malady.
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 unknownσ,
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