Consider the pay-off table below, with three alternatives Aj and three states of nature Sj in $. Let P(S1) = 0.30, P(S2) = 0.50, P(S3) = 0.20. Compute the expected monetary value for each of the alternatives. (Round the final answer to the nearest dollar
mathematics
Description
Assignment #6 (8%)
This assignment relates to the Course Learning
Requirements:
CLR5 Apply decision making using Statistical Decision
Theory
Objective of this Assignment:
Evaluate important concepts in
the theory of decision making, including expected monetary value and expected
opportunity loss.
Instructions:
Use Word or Excel to solve the following exercises. Please show all your work.
1. (9 points) Consider the pay-off table below, with three
alternatives Aj and three states of nature Sj in $. Let
P(S1) = 0.30, P(S2) = 0.50, P(S3) = 0.20.
Compute the expected monetary value for each of the alternatives. (Round the
final answer to the nearest dollar.)
|
Alternative |
S1($) |
S2($) |
S3($) |
|
A1 |
50 |
70 |
100 |
|
A2 |
90 |
40 |
80 |
|
A3 |
70 |
60 |
90 |
2. (4 points) Consider the pay-off table below, with three
alternatives Aj and three states of nature Sj in $. Let
P(S1) = 0.30, P(S2) = 0.50, P(S3) = 0.20.
Develop an opportunity loss table. (Round the final answer to the nearest
dollar.)
Pay-off Table:
|
Alternative |
S1($) |
S2($) |
S3($) |
|
A1 |
50 |
70 |
100 |
|
A2 |
90 |
40 |
80 |
|
A3 |
70 |
60 |
90 |
3. (10 points) Consider the pay-off table below, with three
alternatives Aj and three states of nature Sj in $. Let
P(S1) = 0.30, P(S2) = 0.50, P(S3) = 0.20.
Compute the expected opportunity losses for each of the alternatives. (Round
the final answer to the nearest dollar.)
Pay-off Table:
|
Alternative |
S1($) |
S2($) |
S3($) |
|
A1 |
50 |
70 |
100 |
|
A2 |
90 |
40 |
80 |
|
A3 |
70 |
60 |
90 |
4. (14 points) The Wilhelms Cola Company plans to market a new
pineapple-flavoured cola this summer. The decision is whether to package the
cola in returnable or in non-returnable bottles. Currently, the provincial
legislature is considering eliminating non-returnable bottles. Tybo Wilhelms,
president of Wilhelms Cola Company, has discussed the problem with his
government representative and established the probability to be 0.70 that
non-returnable bottles will be eliminated. The table below shows the estimated
monthly profits (in thousands of dollars) if the cola is bottled in returnable
versus nonreturnable bottles. Of course, if the law is passed and the decision
is to bottle the cola in nonreturnable bottles, all profits would be from
out-of-province sales.
|
Alternative |
Law is
passed (S1) |
Law is
not passed (S2) |
|
Returnable
Bottle |
$80 |
$40 |
|
Non-returnable
Bottle |
$25 |
$60 |
a) Develop an opportunity loss table, and determine the
opportunity loss for each decision.
(Round the final answers to nearest dollar) (8 marks)
b) Compute the expected profit for both bottling decisions.
(Round the final answers to nearest 10th.) (6 marks)
5. (16 points) Blackbeard's Phantom Fireworks is considering
two new bottle rockets. The company can add both to the current line, neither,
or just one of the two. The success of these products depends on consumers' reactions
to the products. These reactions can be summarized as good, P(S1) = 0.30; fair,
P(S2) = 0.50; or poor, P(S3) = 0.20. The company's revenues, in thousands of
dollars, are estimated in the accompanying pay-off
table:
|
Decision |
S1 |
S2 |
S3 |
|
Neither |
$0 |
$0 |
$0 |
|
Product
1 only |
$125 |
$65 |
$30 |
|
Product
2 only |
$105 |
$60 |
$30 |
|
Both |
$220 |
$110 |
$40 |
a)
Compute the expected monetary value for
each decision. (Round the final answers to 2 decimal places.) (4 marks)
b)
What decision would you recommend? (1
mark)
c)
Develop an opportunity loss table.
(Round the final answers to the nearest whole number.) (4 marks)
d)
Compute the expected opportunity loss
for each decision. (Round the final answers to 2 decimal places.) (4 marks)
e)
Compute the expected value of perfect
information. (Round the final answer to the nearest whole number.) (3 marks)
