620 CHAPTER 12 Analysis of Variance Bonferroni Multiple Comparison Test Step 1: Do a separate t test for each pair of samples, but make the adjustments described in the following steps. Step 2: For an estimate of the variance s 2 that is common to all of the involved populations, use the value of MS(error), which uses all of the available sample data. The value of MS(error) is typically obtained with the results when conducting the analysis of variance test. Using the value of MS(error), calculate the value of the test statistic t, as shown below. The particular test statistic calculated below is based on the choice of Sample 1 and Sample 2; change the subscripts and use another pair of samples until all of the different possible pairs of samples have been tested. t = x1 - x2 A MS1error2 # a 1 n1 + 1 n2b Step 3: After calculating the value of the test statistic t for a particular pair of samples, find either the critical t value or the P-value, but make the following adjustment so that the overall significance level does not increase. P-Value Use the test statistic t with df = N - k, where N is the total number of sample values and k is the number of samples, and find the P-value using technology or Table A-3, but adjust the P-value by multiplying it by the number of different possible pairings of two samples. (For example, with three samples, there are three different possible pairings, so adjust the P-value by multiplying it by 3.) Critical Value When finding the critical value, adjust the significance level a by dividing it by the number of different possible pairings of two samples. (For example, with three samples, there are three different possible pairings, so adjust the significance level by dividing it by 3.) Note that in Step 3 of the preceding Bonferroni procedure, either an individual test is conducted with a much lower significance level or the P-value is greatly increased. Rejection of equality of means therefore requires differences that are much farther apart. This adjustment in Step 3 compensates for the fact that we are doing several tests instead of only one test. Example 1 in this section used analysis of variance with the sample data in Table 12-1. We concluded that there is sufficient evidence to warrant rejection of the claim of equal means. Use the Bonferroni test with a 0.05 significance level to identify which mean is significantly different from the others. CP EXAMPLE 2 Bonferroni Test SOLUTION The Bonferroni test requires a separate t test for each of six different possible pair of samples. Here are the null hypotheses to be tested: H0: m1 = m2 H0: m1 = m3 H0: m1 = m4 H0: m2 = m3 H0: m2 = m4 H0: m3 = m4 We begin with H0: m1 = m2. Using the sample data given in Table 12-1, we have n1 = 12 and x1 = 290.0. Also, n2 = 12 and x2 = 180.75. From the technology results shown in Example 1 we also know that MS1error2 = 5026.337121.
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