Large-Sample Approximation in .NET

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Large-Sample Approximation
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When both n1 and n2 are moderately large, say, greater than 8, the distribution of w1 can be well approximated by the normal distribution with mean n1 1n1 n2 2 12
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CHAPTER 15 NONPARAMETRIC STATISTICS
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Therefore, for n1 and n2
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as a statistic, and the appropriate critical region is 0 z0 0 z /2, z0 depending on whether the test is a two-tailed, upper-tail, or lower-tail test.
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Comparison to the t-Test
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In Section 15-3.4 we discussed the comparison of the t-test with the Wilcoxon signed-rank test. The results for the two-sample problem are identical to the one-sample case. That is, when the normality assumption is correct, the Wilcoxon rank-sum test is approximately 95% as ef cient as the t-test in large samples. On the other hand, regardless of the form of the distributions, the Wilcoxon rank-sum test will always be at least 86% as ef cient. The ef ciency of the Wilcoxon test relative to the t-test is usually high if the underlying distribution has heavier tails than the normal, because the behavior of the t-test is very dependent on the sample mean, which is quite unstable in heavy-tailed distributions.
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EXERCISES FOR SECTION 15-4
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15-28. An electrical engineer must design a circuit to deliver the maximum amount of current to a display tube to achieve suf cient image brightness. Within her allowable design constraints, she has developed two candidate circuits and tests prototypes of each. The resulting data (in microamperes) are as follows: Circuit 1: 251, 255, 258, 257, 250, 251, 254, 250, 248 Circuit 2: 250, 253, 249, 256, 259, 252, 260, 251 Use the Wilcoxon rank-sum test to test H0: 1 2 against the alternative H1: 1 > 2. Use 0.025. 15-29. One of the authors travels regularly to Seattle, Washington. He uses either Delta or Alaska. Flight delays are sometimes unavoidable, but he would be willing to give most of his business to the airline with the best on-time arrival record. The number of minutes that his ight arrived late for the last six trips on each airline follows. Is there evidence that either airline has superior on-time arrival performance Use 0.01 and the Wilcoxon rank-sum test. Delta: 13, 10, 1, 4, 0, 9 (minutes late) Alaska: 15, 8, 3, 1, 2, 4 (minutes late) 15-30. The manufacturer of a hot tub is interested in testing two different heating elements for his product. The element that produces the maximum heat gain after 15 minutes would be preferable. He obtains 10 samples of each heating unit and tests each one. The heat gain after 15 minutes (in F) follows. Is there any reason to suspect that one unit is superior to the other Use 0.05 and the Wilcoxon rank-sum test. Unit 1: 25, 27, 29, 31, 30, 26, 24, 32, 33, 38 Unit 2: 31, 33, 32, 35, 34, 29, 38, 35, 37, 30 15-31. Use the normal approximation for the Wilcoxon rank-sum test for the problem in Exercise 15-28. Assume that 0.05. Find the approximate P-value for this test statistic. 15-32. Use the normal approximation for the Wilcoxon rank-sum test for the heat gain experiment in Exercise 15-30. Assume that 0.05. What is the approximate P-value for this test statistic 15-33. Consider the chemical etch rate data in Exercise 10-21. Use the Wilcoxon rank-sum test to investigate the claim that the mean etch rate is the same for both solutions. If 0.05, what are your conclusions 15-34. Use the Wilcoxon rank-sum test for the pipe de ection temperature experiment described in Exercise 10-20. If 0.05, what are your conclusions 15-35. Use the normal approximation for the Wilcoxon rank-sum test for the problem in Exercise 10-21. Assume that 0.05. Find the approximate P-value for this test. 15-36. Use the normal approximation for the Wilcoxon rank-sum test for the problem in Exercise 10-20. Assume that 0.05. Find the approximate P-value for this test.
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