Answer all of the problems below. You may use notes, books, and calculators.
1. Gadget production follows the following Cobb-Douglas production function:
Q = 10K.5L.5, where K is the number of units of capital and L is hours of labor.
Each hour of labor costs $10 and each unit of capital costs $20. If you are budgeted $50,000 per month and gadgets sell for $100 apiece, find the optimal production of gadgets and the monthly profits on gadgets.
2. 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 |
3. Given the following returns for an investment project, in $ thousands, find the optimal investment level using four criteria: maximax, minimax regret, maximin, and maximum expected value. 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 |
|
4. Gidgets cost $100 apiece at wholesale and sell for $250 at retail. As a retailer you have determined that monthly gidget demand is approximately normally-distributed with mean of 1000 and standard deviation of 150 units. Assuming that unbought gadgets cannot be resold, find the optimal number of gadgets to order each month.
5. From the data in the table below, calculate the sample correlation coefficient between the variables X and Y. (The number of columns may be greater than needed)
|
X |
Y |
|
|
|
|
|
|
|
10 |
125 |
|
|
|
|
|
|
|
12 |
140 |
|
|
|
|
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|
|
14 |
145 |
|
|
|
|
|
|
|
16 |
165 |
|
|
|
|
|
|
|
18 |
175 |
|
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6. The data below
are a random sample of five mortgage rates, in percent, in the
7.0, 7.5, 7.75, 7.25, 7.0
a. Calculate the
sample mode, mean, median, range and standard deviation for the data.
b. Construct a 90 percent confidence interval for mortgage rates in the area.
State your assumptions? Do you think these are realistic?
7. The table below
lists the probability distribution for the number of classes missed by Citadel
MBA students.
|
Absences
(X) |
P(X) |
|
|
|
0 |
.4 |
|
|
|
1 |
.3 |
|
|
|
2 |
.2 |
|
|
|
3 |
.1 |
|
|
|
|
|
|
|
a) Find the
expected number of classes missed by each student and the standard deviation.
b) What is the probability that a student,
chosen at random, misses more than 1.4 classes?
c) During a semester
with 100 enrollments in courses, what is the probability that the average
number of classes missed per student will exceed 1.40? [Assume absences are
independent]
8. The probability
that a stock listed on the Over-the-Counter market (OTC) will remain listed on
that market six months later is .8.
a) If a sample of
the names of 25 OTC stocks is chosen at random, what is the probabilility
that 6 or more of these stocks will not be listed on the OTC six months later?
b) From a list of
200 such OTC stocks, what is the probability that between 130 and 150,
inclusive, of these stocks will still be listed on the OTC six months later?
9. When asked how
often a person jumping off the
Supposing that this
meant about one percent of the time,
a) how many survivors would one expect of the 350 people who
have attempted this feat?
b) what is the probability that more than five of these 350
jumpers survived?