Please find attached file for question data. You have to solve questions from 8-11. Thanks

6. Please refer to the following figure in answering the problem.

In the oil-wildcatting problem, suppose that the company could collect information from a drilling core sample and analyze it to determine whether a dome structure exists at Site 1. A positive result would indicate the presence of a dome, and a negative result would indicate the absence of a dome. The test is not perfect, however. The test is highly accurate for detecting a dome; if there is a dome, then the test shows a positive result 99% of the time. On the Other hand if there is no dome, the probability of a negative result is only 0.85. Thus, P(+ | Dome) = 0.99 and P(- | No Dome) = 0.85. Use these probabilities, the information given in the example, and Bayes’ theorem to find the posterior prob abilities P(Dome | +) and P(Dome | —). If the test gives a positive result, which site should be selected? Calculate expected values to support your conclusion! If the test result is negative, which site should be chosen? Again, calculate expected values.

7. In Exercise 6, calculate the probability that the test is positive and a dome structure exists [P(+ and Dome)]. Now calculate the probability of a positive result, a dome structure, and a dry hole [P(+ and Dome and Dry)]. Finally, calculate P(Dome |+ and Dry).

8. Referring to the oil-wildcatting decision diagrammed in Exercise 6, suppose that the decision maker has not yet assessed P(Dome) for Site 1. Find the value of P(Dome) for which the two sites have the same EMV. If the decision maker believes that P(Dome) is somewhere between 0.55 and 0.65, what action should be taken?

9. Again referring to the Figure in Exercise 6, suppose the decision maker has not yet assessed P(Dry) for Site 2 or P(Dome) for Site 1. Let P(Dry) = p and P(Dome) = q. Construct a two-way sensitivity analysis graph for this decision problem.

10. You are the mechanical engineer in charge of maintaining the machines in a factory. The plant manager has asked you to evaluate a proposal to replace the current machines with new ones. The old and new machines perform substantially the same jobs, and so the question is whether the new machines are more reliable than the old. You know from past experience that the old machines break down roughly according to a Poisson distribution, with the expected number of breakdowns at 2.5 per month. When one breaks down, $150 is required to fix it. The new machines, however, have you a bit confused. According to the distributor’s brochure, the new machines are supposed to break down at a rate of 1.5 machines per month on average and should cost $170 to fix. But a friend in another plant that uses the new machines reports that they break down at a rate of approximately 3.0 per month (and do cost $170 to fix). (In either event, the number of breakdowns in any month appears to follow a Poisson distribution.) On the basis of this information, you judge that it is equally likely that the rate is 3.0 or 1.5 per month.

a) Based on minimum expected repair costs, should the new machines be adopted?

b) Now you learn that a third plant in a nearby town has been using these machines. They have experienced 6 breakdowns in 3.0 months. Use this information to find the posterior probability that the breakdown rate is 1.5 per month.

c) Given your posterior probability, should your company adopt the new machines in order to minimize expected repair costs?

11. For the decision tree below, assume Chance Events E and F are independent.

a ) Draw the appropriate decision tree and calculate the EVPI for Chance Event E only.

b ) Draw the appropriate decision tree and calculate the EVPI for Chance Event F only.

c) Draw the appropriate decision tree and calculate the EVPI for both Chance Events E and F: that is, perfect information for both E and F is available before a decision is made.

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