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the Kruskal-Wallis test is approximately equivalent to applying the usual analysis of variance to the ranks. The rank transformation has wide applicability in experimental design problems for which no nonparametric alternative to the analysis of variance exists. If the data are ranked and the ordinary F-test is applied, an approximate procedure results, but one that has good statistical properties. When we are concerned about the normality assumption or the effect of outliers or wild values, we recommend that the usual analysis of variance be performed on both the original data and the ranks. When both procedures give similar results, the analysis of variance assumptions are probably satis ed reasonably well, and the standard analysis is satisfactory. When the two procedures differ, the rank transformation should be preferred since it is less likely to be distorted by nonnormality and unusual observations. In such cases, the experimenter may want to investigate the use of transformations for nonnormality and examine the data and the experimental procedure to determine whether outliers are present and why they have occurred. EXERCISES FOR SECTION 15-5

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15-37. Montgomery (2001) presented the results of an experiment to compare four different mixing techniques on the tensile strength of portland cement. The results are shown in the following table. Is there any indication that mixing technique affects the strength Use 0.05. Mixing Technique 1 2 3 4 3129 3200 2800 2600 de ection angles for ve saws chosen at random from each of four different manufacturers. Is there any evidence that the manufacturers products differ with respect to kickback angle Use 0.01. Manufacturer Tensile Strength (lb/in.2) 3000 3000 2900 2700 2865 2975 2985 2600 2890 3150 3050 2765 A B C D 42 28 57 29 Kickback Angle 17 50 45 40 24 44 48 22 39 32 41 34 43 61 54 30

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15-38. An article in the Quality Control Handbook, 3rd edition (McGraw-Hill, 1962) presents the results of an experiment performed to investigate the effect of three different conditioning methods on the breaking strength of cement briquettes. The data are shown in the following table. Using 0.05, is there any indication that conditioning method affects breaking strength Conditioning Method 1 2 3 553 553 492

15-40. Consider the data in Exercise 13-2. Use the Kruskal-Wallis procedure with 0.05 to test for differences between mean uniformity at the three different gas ow rates. 15-41. Find the approximate P-value for the test statistic computed in Exercise 15-37. 15-42. Find the approximate P-value for the test statistic computed in Exercise 15-40.

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Breaking Strength (lb/in.2) 550 599 530 568 579 528 541 545 510 537 540 571 15-43. The surface nish of 10 metal parts produced in a grinding process is as follows: (in microinches): 10.32, 9.68, 9.92, 10.10, 10.20, 9.87, 10.14, 9.74, 9.80, 10.26. Do the data support the claim that the median value of surface nish is 10 microinches Use the sign test with 0.05. What is the P-value for this test 15-44. Use the normal appoximation for the sign test for the problem in Exercise 15-43. Find the P-value for this test. What are your conclusions if 0.05 15-45. Fluoride emissions (in ppm) from a chemical plant are monitored routinely. The following are 15 observations

15-39. In Statistics for Research (John Wiley & Sons, 1983), S. Dowdy and S. Wearden presented the results of an experiment to measure the performance of hand-held chain saws. The experimenters measured the kickback angle through which the saw is de ected when it begins to cut a 3-inch stock synthetic board. Shown in the following table are

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