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Question
NARRBEGIN: SA_84_87The owner of a popular Internet-based auction site believes that more than half of the people who sell items on her site are women. To test this hypothesis, the owner sampled 1000 customers who sold items on her site and she found that 53% of the customers sampled were women. Some calculations are shown in the table below
| Sample proportion | 0.53 |
| Standard error of sample proportion | 0.01578 |
| Z test statistic | 1.9008 |
| p-value | 0.0287 |
(A) If you were to conduct a hypothesis test to determine if greater than 50% of customers who use this Internet-based site are women, would you conduct a one-tail or a two-tail hypothesis test? Explain your answer, and state the appropriate null and alternative hypotheses.
(B) How many customers out of the 1000 sampled must have been women in this case?
(C) Using a 5% significance level, can the owner of this site conclude that women make up more than 50% of her customers? Explain your answer.
(D) If you were to use a 1% significance level, would the conclusion from (C) change? Explain your answer.
Answer
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NARRBEGIN: SA_55_62A landowner in Texas is offered $200,000 for the exploration rights to oil on her land, along with a 25% royalty on the future profits if oil is discovered. The landowner is also tempted to develop the field herself, believing that the interest in her land is a good indication that oil is present. In that case, she will have to contract a local drilling company to drill an exploratory well on her own. The cost for such a well is $750,000, which is lost forever if no oil is found. If oil is discovered, however, the landowner expects to earn future profits of $7,500,000. Finally, the landowner estimates (with the help of her geologist friend) the probability of finding oil on this site to be 60%.NARREND(A) Construct a decision tree to help the landowner make her decision. Make sure to label all decision and chance nodes and include appropriate costs, payoffs and probabilities.(B) What should the landowner do? Why?(C) Suppose the landowner is uncertain about the reliability of her geologist friend's estimate of the probability that oil will be found on her land. If she thinks the probability could be anywhere between 40% and 80%, would that change her decision?(D) Suppose that, in addition to the uncertainty about the probability of finding oil, the landowner is also uncertain about the cost of the exploratory well (could vary +/- 25%) and the future profits (could vary +/- 50%). To which of these variables is the expected value most sensitive?(E) What does the risk profile show about the relative risk levels for the landowner's two options?(F) Suppose the landowner suspects that she may be a somewhat risk-averse decision maker, because the she doesn"t feel there is as much of a difference between the two options as their expected values would indicate. She consults with a decision analysis expert who asks her to decide between two hypothetical alternatives: 1) a gamble with equal probabilities of winning an amount $X and losing an amount "$X/2, and 2) doing nothing, with a payoff of $0. The point at which she cannot decide between 1) and 2) is when X=$1,500,000. What is her risk tolerance if she uses an exponential utility function to model her preferences?(G) Apply the risk tolerance given in your answer to the previous question to the landowner's decision tree in (A). What is the optimal decision in this case? What is the resulting certainty equivalent?(H) If the landowner could hire an expert geologist prepare a report to help her make her decision, what is the most that information could be worth? Assume the geologist's information is perfectly reliable.
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