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Question
NARRBEGIN: SA_96_100Q-Mart is interested in comparing customers who used its own charge card with those who use other types of credit cards. Q-Mart would like to know if customers who use the Q-Mart card spend more money per visit, on average, than customers who use some other type of credit card. They have collected information on a random sample of 38 charge customers and the data is presented below. On average, the person using a Q-Mart card spends $192.81 per visit and customers using another type of card spend $104.47 per visit.
Summary statistics for two samples
| Q-Mart | Other Charges | |
| Sample sizes | 13 | 25 |
| Sample means | 192.81 | 104.47 |
| Sample standard deviations | 115.243 | 71.139 |
| Sample mean difference | 88.34 |
| Pooled standard deviation | 88.323 |
| Std error of difference | 30.201 |
| t-test statistic | 2.925 |
| p-value | 0.006 |
(A) Given the information above, what is
and 
for this comparison? Also, does this represent a one-tailed or a two-tailed test? Explain your answer.(B) What are the degrees of freedom for the t-statistic in this calculation? Explain how you would calculate the degrees of freedom in this case.
(C) What is the assumption in this case that allows you to use the pooled standard deviation for this test?
(D) Using a 5% level of significance, is there sufficient evidence for Q-Mart to conclude that customers who use the Q-Mart card charge, on average, more than those who use another charge card? Explain your answer.
(E) Using a 1% level of significance, is there sufficient evidence for Q-Mart to conclude that customers who use the Q-Mart card charge, on average, more than those who use another charge card? 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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