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The null hypothesis to be tested is that the K population means are equal-that is, HO:M1 = M2 = ... = MK We make the following additional assumptions: V13 1. The population variances are equal. 2. The population distributions are normal. A test of significance level a is provided by the decision rule MSG reject H, if MSW > FK-1n-Ka where FK-11-K, is the number for which P(FK-18-K > FK-1,-K.x) = a and the random variable FK-1,2-K follows an F distribution with numerator de- grees of freedom (K-1) and denominator degrees of freedom (n - K). The p-value for this test is the smallest significance value that would allow us to reject the null hypothesis. For the fuel consumption data, we find the following: MSG 10.775 = 15.039 0.71647 MSW The numerator and denominator degrees of freedom are, respectively, (K-1) = 2 and (n-K) = 17. Thus, for a 1% significance level test, from Appendix Table 9, we have the following: F2,17,0.01 = 6.112 Hence, these data allow us to reject, at the 1% significance level, the null hypothesis that population mean fuel consumption is the same for all three types of automobiles. The computations involved in carrying out this test are very conveniently summarized in a one-way analysis of variance table. The general form of the table is set out in Table 15.3. For the fuel consumption data the analysis of variance is set out in Table 15.4. Note that in some expositions the within-groups sum of squares is referred to as the error sum of squares. SOURCE OF VARIATION SUM OF SQUARES DEGREES OF FREEDOM MEAN SQUARES F RATIO + MSG Between groups SSG K-1 SSG MSG = K-1 MSW SSW Within groups SSW n - K MSW = n-K Total SST n-1 SOURCE OF VARIATION SUM OF SQUARES DEGREES OF FREEDOM MEAN SQUARES F RATIO r Between groups 21.55 2 10.78 15.05 Within groups 12.18 17 0.7165 Total 33.73 19 Example 15.1 Reading Difficulty of Magazine Advertisements (One-Way Analysis of Variance) The fog index is used to measure the reading difficulty of a written text: The higher the value of the index, the more difficult the reading level. We want to know if the reading difficulty index is different for three magazines: Scientific American, Fortune, and the New Yorker