13-5 Kruskal-Wallis Test for Three or More Samples 673 INTERPRETATION There is sufficient evidence to reject the claim that the three samples of HIC measurements come from populations with medians that are all equal. At least one of the population medians appears to be different from the others. Head Injuries in Small, Midsize, and Large Cars EXAMPLE 1 Table 13-6 lists head injury criterion (HIC) measurements of small, midsize, and large car crash tests. Use a 0.05 significance level to test the claim that the three samples of HIC measurements are from populations with medians that are all equal. REQUIREMENT CHECK (1) Each of the three samples is a simple random independent sample. (2) Each sample size is at least 5. The requirements are satisfied. The null and alternative hypotheses are as follows: H0: The populations of small cars, midsize cars, and large cars all have the same median HIC measurement in crash tests. H1: The three populations of small, midsize, and large cars have median HIC measurements that are not all the same. Test Statistic First combine all of the sample data and rank them, then find the sum of the ranks for each category. In Table 13-6, ranks are shown in parentheses next to the original sample values. Next, find the sample size (n) and sum of ranks (R) for each sample. Those values are shown at the bottom of Table 13-6. Because the total number of observations is 19, we have N = 19. We can now evaluate the test statistic as follows: H = 12 N1N + 12 a R2 1 n1 + R2 2 n2 + g + R2 k nk b - 31N + 12 = 12 19119 + 12 a 1102 8 + 47.52 6 + 32.52 5 b - 3119 + 12 = 6.309 Because each sample has at least five observations, the distribution of H is approximately a chi-square distribution with k - 1 degrees of freedom. The number of samples is k = 3, so we have 3 - 1 = 2 degrees of freedom. P-Value With H = 6.309 and df = 2, Table A-4 shows that the P-value is less than 0.05. Using technology, we get P@value = 0.043. Because the P-value is less than the significance level of 0.05, we reject the null hypothesis of equal population medians. Critical Value Refer to Table A-4 to find the critical value of 5.991, which corresponds to 2 degrees of freedom and a 0.05 significance level (with an area of 0.05 in the right tail). In this right-tailed test with test statistic H = 6.309 and critical value of 5.991, the test statistic does exceed the critical value, so it does fall within the critical region. We reject the null hypothesis of equal population medians. YOUR TURN. Do Exercise 5 “HIC Measurements.”
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