Answer
both of the problems below. You may use
notes, books, and calculators.
1. Find the optimal
strategies for the two players and the expected value of the following
two-person, zero-sum game.
A's Payoff Matrix
|
|
B1 |
B2 |
|
A1 |
8 |
-5 |
|
A2 |
-4 |
10 |
2. Given the following returns for an investment
project, in $ thousands, find the optimal investment level using four
criteria: maximax,
minimax regret, maximin,
and the maximum expected value; calculate the standard deviations for the three
investment levels, as well. Also calculate the expected value of perfect
information.
State of
|
Investment Level |
Good |
So-so |
Bad |
Expected Value |
|
High |
400 |
250 |
-400 |
|
|
Medium |
300 |
100 |
-100 |
|
|
Low (zero) |
0 |
0 |
0 |
|
|
Probabilities |
.2 |
.5 |
.3 |
|
3. A mid-western city is
trying to determine how large an arena to build for its new hockey team. A consultant has estimated that demand for
tickets, that is, seats, for each game at the expected price of $15 is
normally-distributed with a mean of 12,000 and standard deviation of 5000. According to the engineering contractor building
the arena the cost per seat per game for the expected life of the arena is
estimated at $5. How many seats should
the arena have?
Statistics
Test.
4. The
data below are a random sample of mortgage rates, in percent, in the
Washington, D.C. metropolitan area for the week of
5.50 5.75 6.25
5.75 5.8
a. Calculate the sample
mean, mode, median, range and standard deviation for the data.
b. Another sample of 40
rates revealed a sample mean of 5.85 and a standard deviation of .32. Construct a 90 percent confidence interval
for mortgage rates in the area.
c. If the true mean mortgage rate and standard deviation are 5.8
and .30, respectively, what can be said about the proportion of mortgage rates
in excess of 6.25 percent if
i)
nothing is
known about the distribution of mortgage rates?
ii) the distribution of
mortgage rates is mound-shaped?
5. The table below lists the probability distribution for the
number of absences of Citadel cadets in my production ops class during a
semester:
|
Absences (X) |
P(X) |
|
|
|
|
0 |
.2 |
|
|
|
|
1 |
.3 |
|
|
|
|
2 |
.3 |
|
|
|
|
3 |
.1 |
|
|
|
|
4 |
.1 |
|
|
|
a. Find the expected number of classes missed by a student and
the standard deviation.
b. What is the probability that a student, chosen at random, will
miss more than two classes?
c. During
a semester with 60 enrollments in two sections, what is the probability that
the sample mean of the number of classes missed will exceed 2? [Assume that my students this semester
constitute a random sample]
6. The probability that a part is defective is .01.
a. Find the probability of exactly one
defect in twenty parts tested at random.
b. Find the probability of more than six
defects in a random sample of 360 parts.
c. What is the
probability that between 15 and 25 defects in a random sample of 1850 parts tested?