Data Models and Decisions

Final Examination 15.063 Communicating with Data

Summer 2003

Problem 1 (30 points)

Joe Bloke operates a garage in Rockwell, MA. He has just been offered the opportunity to install a car wash at the highly discounted cost of $20,000. This exceptional offer, however, is only available if he accepts it within one week. If he does, he will charge $13.00 per car washed. Labor and materials will cost $3.00 per car washed. Joe expects that monthly demand for car wash will be about 100 cars. A major uncertainty is whether or not his competitor, Meg’s Garage and Auto Repair, will also install her own car wash. If Meg does, then Joe expects the demand for his car wash will be halved. Joe figures that there is a 20% chance that Meg will install her car wash very soon, in which case Joe and Meg would split demand for the next 12 months; a 30% chance that Meg will install her car wash in about 6 months, in which case Joe would have the full demand for 6 months and split it with her afterwards; and a 50% chance that Meg will not install a car wash at all. Furthermore, Joe’s accountant estimates that the value of the car wash after the first 12 months is $15,000 in the absence of competition, and $10,000 if Meg has installed her own car wash.

1. (12 points) Construct a decision tree to represent Joe’s problem. Be sure to properly label all branches, and to include all relevant probabilities and payoffs.

1. (5 points) Find the best strategy for Joe, assuming the Expected Monetary Value (EMV) criterion. What is the EMV of this strategy?

1. (5 points) Is EMV an appropriate decision criterion for Joe in such a situation? Why, or why not? (Please limit your answer to no more than 4 lines of text.)

1. (8 points) If the monthly demand was smaller or bigger than 100, would your recommendation change? For which values of monthly demand should Joe choose to have the car wash installed? Assume that he uses the EMV criterion and that all other data are as originally stated.

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Problem 2 (10 points)

Suppose that, at last, Joe opened the car wash referenced in Problem 1 and Meg did not. Now, Joe has to decide the price p that he will charge his customers for washing their cars because he is not sure if $13.00 is the right one. In order to do that, he improved the model he had before (described in Problem 1). Namely, for a certain price p, he assumes that the monthly demand is a normal random variable with mean 100 – p and constant standard deviation equal to 10. To provide a good quality of service he needs to hire one employee if the expected monthly demand is greater than 80, but otherwise he can manage by himself. The monthly salary that the employee makes is a random variable that is uniformly distributed between $800 and $2,000 (that is in addition to the $3 labor and materials mentioned before).

How would you set up a simulation to determine how much Joe should charge (i.e., p) ?

Hint: you do not need to do any math in this problem, just describe briefly how to use simulation to solve the problem.

Problem 3 (10 points)

The CWD-Mart department store has sampled 225 sales records on July 14.

The average sale was $42.00, with an observed sample standard deviation of $30.00 per customer.

1. (5 points) Construct a 90% confidence interval for the mean sale value.

1. (5 points) How many sales records would need to be sampled for the 90% confidence interval to be within ρ$1.00 of the sample mean?

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Problem 4 (30 points)

GuardWare Inc. has installed intrusion-detection devices on the campus of a large university. The devices are very sensitive and, on any given weekend, each one has a 10% chance of being mistakenly activated when no intruder is present. Assume that the mistaken activation of different devices are independent events.

1. There are six devices in the Administration building. Assume there will be no intruder in that building next weekend.

1. (6 points) What is the probability that no device will be activated next weekend?

1. (6 points) If two or more of the six devices in the Administration building are activated during the same weekend, the system automatically signals the police. What is the probability that this system will signal the police next weekend?

Hint: remember the assumption.

1. (6 points) What are the expected value and the standard deviation of the number of devices (mistakenly) activated in the Administration building next weekend?

1. (6 points) There are a total of 200 such devices on campus. Assume there will be no intruder on any of these locations next weekend. What is the probability that at least

1. of these devices will be (mistakenly) activated? (Hint: use an appropriate approximation, and explain why you may do so.)

1. (6 points) An intruder has a 5% chance of not activating the device located in the President’s office. Campus police has obtained information about a student prank in preparation and, as a result, estimates that there is a 20% chance that an intruder will visit the President’s office during next Columbus Day weekend. If this device is activated during that weekend, what is the probability that there is an intruder in the President’s office?

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Problem 5 (20 points)

CWD-Tel operates mobile telephone systems in a developing country. David, a manager at CWD-Tel, wants to predict the peak demand for circuits, on the basis of the number of portable phones in a region. He has obtained the following data for 20 regions:

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