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# Mathematical Statistics with Applications, Seventh Edition

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www.downloadslide.com 13.3 Comparison of More Than Two Means: Analysis of Variance for a One-Way Layout 669 equal to where SSE = = k∑ ∑n i (Y ij − Y i• ) 2 i=1 j=1 k∑ (n i − 1)Si 2 , i=1 Si 2 = 1 ∑n i (Y ij − Y i• ) 2 . n i − 1 Notice that SSE is a function of only the sample variances Si 2 , for i = 1, 2,...,k. Because each of the Si 2 values provides an unbiased estimator for σi 2 = σ 2 with n i − 1 df, an unbiased estimator of σ 2 based on (n 1 + n 2 +···+n k − k) = n − k df is provided by j=1 S 2 = MSE = SSE (n 1 − 1) + (n 2 − 1) +···+(n k − 1) = SSE n − k . Because Y = 1 k∑ ∑n i Y ij = 1 k∑ n i Y i• , n n i=1 j=1 i=1 it follows that SST is a function of only the sample means Y i• , for i = 1, 2,...,k. The MST possesses (k − 1) df—that is, 1 less than the number of means—and is MST = SST k − 1 . To test the null hypothesis, H 0 : µ 1 = µ 2 =···=µ k , against the alternative that at least one of the equalities does not hold, we compare MST with MSE, using the F statistic based on ν 1 = k−1 and ν 2 = n−k numerator and denominator degrees of freedom, respectively. The null hypothesis will be rejected if F = MST MSE > F α, where F α is the critical value of F for a test of level α. In Exercise 13.6, you will prove that, under H 0 : µ 1 = µ 2 =···=µ k , the statistic F possesses an F distribution with k − 1 and n − k numerator and denominator degrees of freedom, respectively.

670 Chapter 13 The Analysis of Variance www.downloadslide.com In keeping with our previous conventions, we will use the notation y ij to denote the observed value of Y ij . Similarly, we will use y i• and y i• to denote the observed values of Y i• and Y i• , for i = 1, 2,...,k, respectively. Intuitively, the greater the differences among the observed values of the treatment means, y 1• , y 2• ,...,y k• , the greater is the evidence to indicate a difference among the corresponding population means. If all of the treatment means are identical, y 1• = y 2• = ··· = y k• = y, and all of the differences that appear in the preceding expression for SST equal zero, implying that SST = 0. As the treatment means get farther apart, the deviations (y i• − y) increase in absolute value and the observed value of SST increases in magnitude. Consequently, the larger the observed value of SST, the greater is the weight of evidence favoring rejection of the null hypothesis. This same line of reasoning applies to the F tests employed in the ANOVA for all designed experiments. The assumptions underlying the ANOVA F tests deserve particular attention. Independent random samples are assumed to have been selected from the k populations. The k populations are assumed to be normally distributed with variances σ 2 1 = σ 2 2 = ··· = σ 2 k = σ 2 and means µ 1 ,µ 2 ,...,µ k . Moderate departures from these assumptions will not seriously affect the properties of the test. This is particularly true of the normality assumption. The assumption of equal population variances is less critical if the sizes of the samples from the respective populations are all equal (n 1 = n 2 = ··· = n k ). A one-way layout with equal numbers of observations per treatment is said to be balanced. EXAMPLE 13.2 Solution Four groups of students were subjected to different teaching techniques and tested at the end of a specified period of time. As a result of dropouts from the experimental groups (due to sickness, transfer, etc.), the number of students varied from group to group. Do the data shown in Table 13.2 present sufficient evidence to indicate a difference in mean achievement for the four teaching techniques? The observed values of the quantities necessary to compute the value of the F statistic are ( ) CM = 1 4∑ n 2 ∑ i y ij = (1779)2 = 137,601.8, n 23 i=1 j=1 Total SS = 4∑ ∑n i i=1 yij 2 j=1 − CM = 139,511 − 137,601.8 = 1909.2, 4∑ yi• 2 SST = − CM = 138,314.4 − 137,601.8 = 712.6, n i=1 i SSE = Total SS − SST = 1196.6. The observed values of MST and MSE are MST = SST k − 1 = 712.6 = 237.5, 3 MSE = SSE n − k = 1196.6 = 63.0. 19

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