12-1 One-Way ANOVA 617 PART 2 Calculations and Identifying Means That Are Different Calculating the Test Statistic F with Equal Sample Sizes n Table 12-2 can be very helpful in understanding the methods of ANOVA. In Table 12-2, compare Data Set A to Data Set B to see that Data Set A is the same as Data Set B with this notable exception: The Sample 1 values each differ by 10. If the data sets all have the same sample size (as in n = 4 for Table 12-2), the following calculations aren’t too difficult, as shown here. TABLE 12-2 Effect of a Mean on the F Test Statistic Add 10 to data in Sample 1 Data Set A Data Set B Sample 1 Sample 2 Sample 3 Sample 1 Sample 2 Sample 3 7 6 4 17 6 4 3 5 7 13 5 7 6 5 6 16 5 6 6 8 7 16 8 7 n1 = 4 n2 = 4 n3 = 4 n1 = 4 n2 = 4 n3 = 4 x1 = 5.5 x2 = 6.0 x3 = 6.0 x1 = 15.5 x2 = 6.0 x3 = 6.0 s2 1 = 3.0 s 2 2 = 2.0 s 2 3 = 2.0 s 2 1 = 3.0 s 2 2 = 2.0 s 2 3 = 2.0 Data Set A Data Set B Step 1: Variance between samples nsx 2 = 4(0.0833) = 0.3332 ns x 2 = 4(30.0833) = 120.3332 Step 2: Variance within samples sp 2 = 3.0 + 2.0 + 2.0 3 = 2.3333 sp 2 = 3.0 + 2.0 + 2.0 3 = 2.3333 Step 3: F test statistic F = ns x 2 s 2 p = 0.3332 2.3333 = 0.1428 F = ns x 2 s 2 p = 120.3332 2.3333 = 51.5721 P-value P@value = 0.8688 P@value = 0.0000118 Step 1: Find the Variance Between Samples Calculate the variance between samples by evaluating nsx 2 where s x 2 is the variance of the sample means and n is the size of each of the samples. That is, consider the sample means to be an ordinary set of values and calculate the variance. (From the central limit theorem, sx = s>1n can be solved for s to get s = 1n # sx, so that we can estimate s 2 with ns x 2.) For example, the sample means for Data Set A in Table 12-2 are 5.5, 6.0, and 6.0, and these three values have a variance of sx 2 = 0.0833, so that variance between samples = nsx 2 = 410.08332 = 0.3332 Step 2: Find the Variance Within Samples Estimate the variance within samples by calculating sp 2, which is the pooled variance obtained by finding the mean of the sample variances. The sample variances in Table 12-2 are 3.0, 2.0, and 2.0, so that variance within samples = sp 2 = 3.0 + 2.0 + 2.0 3 = 2.3333 continued
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