618 CHAPTER 12 Analysis of Variance Step 3: Calculate the Test Statistic Evaluate the F test statistic as follows: F = variance between samples variance within samples = nsx 2 s 2 p = 0.3332 2.3333 = 0.1428 Finding the Critical Value The critical value of F is found by assuming a right-tailed test because large values of F correspond to significant differences among means. With k samples each having n values, the numbers of degrees of freedom are as follows. Degrees of Freedom 1using k = number of samples and n = sample size2 Numerator degrees of freedom = k - 1 Denominator degrees of freedom = k1n - 12 For Data Set A in Table 12-2, k = 3 and n = 4, so the degrees of freedom are 2 for the numerator and 314 - 12 = 9 for the denominator. With a = 0.05, 2 degrees of freedom for the numerator, and 9 degrees of freedom for the denominator, the critical F value from Table A-5 is 4.2565. If we were to use the critical value method of hypothesis testing with Data Set A in Table 12-2, we would see that this right-tailed test has a test statistic of F = 0.1428 and a critical value of F = 4.2565, so the test statistic is not in the critical region. We therefore fail to reject the null hypothesis of equal means. Understanding the Effect of a Mean on the F Test Statistic To really understand how the method of analysis of variance works, consider Data Set A and Data Set B in Table 12-2 and note the following. ■ The three samples in Data Set A are identical to the three samples in Data Set B, except for this: Each value in Sample 1 of Data Set B is 10 more than the corresponding value in Data Set A. ■ Adding 10 to each data value in the first sample of Data Set A has a significant effect on the test statistic, with F changing from 0.1428 to 51.5721. ■ Adding 10 to each data value in the first sample of Data Set A has a dramatic effect on the P-value, which changes from 0.8688 (not significant) to 0.0000118 (significant). ■ The three sample means in Data Set A (5.5, 6.0, 6.0) are very close, but the sample means in Data Set B (15.5, 6.0, 6.0) are not close. ■ The three sample variances in Data Set A are identical to those in Data Set B. ■ The variance between samples in Data Set A is 0.3332, but for Data Set B it is 120.3332 (indicating that the sample means in B are farther apart). ■ The variance within samples is 2.3333 in both Data Set A and Data Set B, because the variance within a sample isn’t affected when we add a constant to every sample value. The change in the F test statistic and the P-value is attributable only to the change in x1. This illustrates the key point underlying the method of one-way analysis of variance: The F test statistic is very sensitive to sample means, even though it is obtained through two different estimates of the common population variance.
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