### MATH 318: Spring 2008 Sample Final

The final exam for this class is open book/open note. You may use a calculator, ruler, pencil, pen and graph paper on the final. You may not use a cell phone calculator, computer, or any other device with the ability to communicate outside the classroom.  Record all answers and the work needed to justify them in your bluebook.

How is this sample final related to the actual final exam? Before the last week of class I created a list of final exam questions. Four of those questions are on the sample final and four are on the final exam. Some questions on the final exam will be related to questions on the sample final (e.g. the same question appears on the final with some numbers changed or some information appears on both exams). The average question on the final exam is likely to be easier than the average question on the sample final. Do not expect the questions on the final exam to be of the same type or to cover exactly the same topics as the questions on the sample final.

1. (Problem 6, p. 87) The American Housing Survey reported the following data on the number of bedrooms in rented houses in central cities.  Compute the expected value and variance for number of bedrooms in rented houses.
2.  No. of Bedrooms 0 1 2 3 4 or more No. of Houses(thousands) 547 5012 6100 2644 557

The expected value is the weighted sum of probabilities of the outcomes. Below is a table showing the necessary calculations:

 x f(x) x*f(x) x-μ (x-μ)2 f(x)*(x-μ)2 0 0.037 0 -1.842 3.392 0.125 1 0.337 0.337 -0.842 0.709 0.239 2 0.410 0.821 0.158 0.025 0.010 3 0.178 0.534 1.158 1.341 0.239 4 0.037 0.150 2.158 4.567 0.175

The sum of the third column gives the expected value: μ = 1.84.

The sum of the sixth column gives the variance: 0.787.

To quickly check your work, remember that the expected value is a sort of average -- your answer should be between 0 and 4. The square root of the variance is the standard deviation; adding the standard deviation to the average (or subtracting it from the average) should give you a result between 0 and 4.

3. The time (in minutes) between telephone calls at an insurance claims office has the exponential probability distribution f(x) = 0.50 e-0.50x for x ≥ 0.

a) What is the probability of 30 seconds or less between telephone calls?

First, convert 30 seconds to .5 minutes. Then compare to the equation on page 83 to see that the mean time between phone calls is μ=2. Using the formula on page 84 we get: P(x ≤ .5) = 1 - e-.5/2 = .2212 (approximately). So the answer is "Approximately 22%."

4. Shown below is a crosstabulation of which Merry Melodies programmers received bonus pay, organized by gender.

 Bonus No Bonus Female 3 1 Male 7 2

a) Create a joint probability table for programmer bonuses.

 Bonus No Bonus Female 23% 8% 31% Male 54% 15% 69% 77% 23% 100%

b) If a programmer got bonus pay, what is the probability that that programmer was female?

This is a conditional probability.

P(female | bonus) = P(female and bonus) / P(bonus) = 23/77 = 30%

No, because 31% of all employees are female. So if there was no bias you would expect about 31% of the bonuses to go to women, which is excactly what happened.

5. Figure 4.17 on page 140 of your text shows a decision tree for the Dante Development Corporation.

a) What is the optimal decision strategy for Dante?

To generate a decision strategy, first fill in the expected values of each chance and decision event starting at the right and moving left. For instance, if market research predicts high demand (chance node 8), the expected value of building a complex is:

0.85 * 2650 + 0.15 * 650 = 2350

The expected value of selling the rights is only 1150, so the value of the decision node labeled 5 is 2350. (From this node you have the option of choosing an expected profit of 2350 or a known profit of 1150; the expected value approach recommends choosing 2350.)

Find the EV of node 6 using the EV of node 9, then use the EV's of nodes 5 and 6 to compute the EV of node 4:

0.6 * 2350 + 0.4 * 1150 = 1870

It turns out that the expected value of building a complex without doing market research is 2000 (node 7), so at decision node 3 your decision strategy should recommend that choice.

The expected value of bidding then turns out to be 1560, which is better than the expected value of not bidding. So your final answer would be:

"Dante Development Corporation should bid on the contract. If they win, they should proceed to build the complex."

b) Develop a risk profile for this decision strategy.

The possible payoffs for this strategy are:

Lose contract: -200
Build with high demand: 2800
Bulid with moderate demand: 800

The probability of the -200 payoff is 20%.

The probability of winning the contract is 80%, and once this happens there is a 60% chance of high demand and a 40% chance of moderate demand. So overall there is a 0.80 * 0.60 = 0.48 or 48% chance of high demand and a 32% chance of moderate demand.

The risk profile will have three vertical bars, one at x=-200, one at x=800 and one at x=2800 (please be careful that your graph is to scale). The vertical bars will have heights 0.20, 0.32 and 0.48 respectively.

c) If Dante wins the contract but does no market research, does the MiniMax Regret approach recommend building or selling?

The payoff table is as follows:

 High Moderate Build 2800 800 Sell 1300 1300

The regret table looks like:

 High Moderate Build 0 500 Sell 1500 0

The maximum regret occurs when selling the rights, so minimax regret recommends building.

6. Merry Melodies produces two software products: MusicMaker and Computer Cacaphony. They are trying to decide how many copies of each program to produce for the upcoming holiday season.
• Each box of MusicMaker software sold yields \$10 profit. Each Computer Cacaphony sold yields \$6 profit.
• Merry Melodies has enough space in a warehouse for 50 palettes of software. A palette holds 40 copies of MusicMaker software or 100 copies of Computer Cacaphony.
• Projected demand is for 1800 copies of Merry Melodies and 1500 copies of Computer Cacaphony.
• Merry Melodies' lawyers recommend that at least 40% of the software produced be Computer Cacaphony in order to avoid an antitrust suit.

a) Formulate a linear programming problem to decide how many of each to produce.

Variables:
M = # Music Maker produced
C = # Computer Cacaphony produced

Maximize: 10M + 6C

Subject to:
M/40 + C/100 ≤ 50
M ≤ 1800
C ≤ 1500
C .4(M + C)
M ≤ 0
C ≤ 0

b) Solve that linear programming problem and give the amounts of each type of software Merry Melodies should produce.

The feasible region is the upper left quadrilateral above (marked). The extreme points of the feasible region are (0, 1500), (1400, 1500), (approximately) (1579, 1052), and (0,0). Plug these values for (M, C) in to 10M + 6C to find that the maximum profit occurs at M=1400, C=1500. So the answer is "Produce 1400 copies of Music Maker and 1500 copies of Computer Cacaphony."

7. (Problem 17, p. 548) The Porsche Shop is preparing an estimate for the repair of a 1964 model 356SC Porsche. The parts needed cost \$8000. Labor cost is \$400/day for the four tasks shown below:

 Activity Optimistic Most Probable Pessimistic A 3 4 8 B 5 8 11 C 2 4 6 D 4 5 12

Tasks B and C can both be started as soon as A completes, but task D cannot start until B and C are both completed.

a) Develop a project network.

First compute the expected times using the formula:

t = (a + 4m + b)/6

 Activity Optimistic Most Probable Pessimistic Expected Variance A 3 4 8 4.5 0.6944 B 5 8 11 8 1 C 2 4 6 4 0.4444 D 4 5 12 6 1.7778

B|4.5 12.5
8|4.5 12.5
/       \
A  |0 4.5      D|12.5 18.5
4.5|0 4.5      6|12.5 18.5
\       /
C|4.5  8.5
4|8.5 12.5

b) What is the expected project completion time?

18.5 days.

c) If the shop obtains the job with a bid that is based on the expected completion time, what is the probability that it will lose money on the job?

The bid is assumed to be equal to parts costs plus labor costs:

\$8000 + \$400/day * 18.5 days = \$15400

d) What bid should the shop make if they wish to have a 90% chance of making a proft?

At this point you need the variance shown in the table above and computed using the formula:

σ2 = ( (b - a)/6 )2.

As we can see from the network, the critical tasks are A, B and D. The variance for the entire project is therefore approximately:

0.6944 + 1 + 1.7778 = 3.47.

We have a normal distribution with a mean of 18.5 and a standard deviation of 1.86. We want a 90% chance of making a profit, so we need to know a number of days at which we are 90% sure the project is completed. In other words, at what point on the x-axis is the area under the curve 90%? To solve this using a TI 83 Calculator, you would enter InvNorm(.90, 18.5, 1.86).

To solve this using a Z-score you would look up the area to the right of the mean (0.9 - 0.5 = 0.4) in a table of z-scores to get the z-score 1.28. You would then solve the equation:

1.28 = (x - 18.5) / 1.86

for x, to get x = 20.88 days.

We can now calculate a bid based on this number:

\$8000 + \$400/day * 20.88 days = \$16352.

Note: We could have rounded up from 20.88 to 21 and bid \$16,400. However, if we do that and another shop submits a bid for \$16,352 we might lose the contract. On the other hand, rounding down would mean accepting a greater than 10% chance of taking a loss.