Exercises

43. It is hypothesized that the more expert a group of people examining a corporation's financial report, the more variable will be their predictions about its future. Independent random samples, each of thirty individuals, from groups of different levels of expertise were chosen.39 The "low-expertise" group consisted of people who had just completed their first intermediate accounting course. Members of the "high-expertise" group had completed undergraduate studies and were employed by reputable C.P.A. firms. The sample members were asked to predict the next period's operating net cash flow of a company on the basis of its annual financial report. For the low-expertise group, the sample variance was 451.770, while for the high-expertise group it was 1,614.208. Test the null hypothesis that the two population variances are equal against the alternative that the true variance is higher for the high-expertise group.

44. It is hypothesized that the market share of a corporation should vary more in an industry with active price competition than in one with duopoly and tacit collusion. In a study of the steam turbine generator industry/1 it was found that in 4 years of active price competition,

3S However, the tests for equality of population means based on independent samples, discussed in Section 9.6, become rather less robust when the two sample sizes are not equal.

39 D. Snowball, "Some effects of accounting expertise and information load. An empirical study," Accounting, Orgctnizations and Society, 5 (1980), 323-38.

40 B. T. Allen, "Tacit collusion and market sharing: The case of steam turbine generators," Industrial Organization Review, 4 (1976), 48-57.

the variance of General Electric's market share was 114,09. In the following 7 years, in which there was duopoly and tacit collusion, this variance was 16.08. Assume that the data can be regarded as an independent random sample from two normal distributions. Test at the 5%-level the null hypothesis that the two population variances are equal against the alternative that the variance of market share is higher in years of active price competition.

45. In Exercise 34, it was assumed that population variances for assessments of the chance of material fraud were the same for auditors using a red flags questionnaire as for auditors not using this questionnaire. Test this assumption against a two-sided alternative hypothesis.

46. In Exercise 36. it was assumed that population variances were equal for first-year sales of textbooks with plain and expensive cover designs. Test this assumption against a two-sided alternative.

47. In Example 9.8. we tested the hypothesis that the mean numbers of ideas generated were the same for groups with and without a moderator. This lest was based on the assumption that the two population variances were equal. Test this assumption against the alternative that the population variance is higher for groups with a moderator.

48. Refer to Exercise 2. Find the power of a 10%-level test when the true mean lifetime of batteries is 49 hours.

49. Refer to Exercise 3(a). Find the probability of a 5%-level test rejecting the null hypothesis when the true mean impurity concentration is 3.10%.

50. Refer to Exercise 5. Find the probability of a 1%-Ievel test accepting the null hypothesis when the true mean response is 3.95.

51. Refer to Example 9.6. Find the power of a 10%' level test if in fact 45%' of supermarket shoppers are able to state the correct price of an item immediately after putting it into the cart.

52. Refer to Exercise 22. Find the probability of rejecting the null hypothesis with a 5% level test if in fact 20% of all U.S. adults would disagree with the statement.

53. Refer to Exercise 24. Find the probability of accepting the null hypothesis with a 10% level test if in fact 60% of all audit partners agree that cash flow from operations is a valid measure of profitability.

54. A fast-food chain tests cach day that the average weight of its "two-pounders" is at least 32 ounces. The alternative hypothesis is that the average weight is less than 32 ounces, indicating that new processing procedures are needed. The weights of two-pounders can be assumed to be normally distributed, with a standard deviation of 3 ounces. The decision rule adopted is to reject the null hypothesis if the sample mean weight is less than 30.8 ounces.

(a) If random samples of n = 36 two-pounders are selected, what is the probability of a Type I error, using this decision rule?

(b) If random samples of n — 9 two-pounders are selected, what is the probability of a Type I error, using this decision rule? Explain why your answer differs from that in part (a).

(c) Suppose that the true mean weight is 31 ounces. If random samples of thirty-six two-pounders are selected, what is the probability of a Type II error, using this decision rule?

55. A wine producer claims that the proportion of its customers who cannot distinguish its product from frozen grape juice is at most .10. The producer decides to test this null hypothesis against the alternative that the true proportion is more than .10. The decision rule adopted is to reject the null hypothesis if the sample proportion who cannot distinguish between these two flavors exceeds .14.

(a) If a random sample of 100 customers is chosen, what is the probability of a Type I error, using this decision rule?

(b) If a random sample of 400 customers is selected, what is the probability of a Type I error, using this decision rule? Explain, in words and graphically, why your answer differs from that in part (a).

(c) Suppose that the true proportion of customers who cannot distinguish between these flavors is .20. If a random sample of 100 customers is selected, what is the probability of a Type II error?

(d) Suppose that instead of the given decision rule, it is decided to reject the null hypothesis if the sample proportion of customers who cannot distinguish between the two flavors exceeds .16. A random sample of 100 customers is selected.

(i) Without doing the calculations, state whether the probability of a Type I error will be higher than, lower than, or the same as that in part (a).

(ii) If the true proportion is .20, will the probability of a Type II error be higher than, lower than, or the same as that in part (c)?

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