An Illustrative Numerical Example for ANOVA-ECO6416

An Illustrative Numerical Example for ANOVA-ECO6416 - An...

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An Illustrative Numerical Example for ANOVA Consider the following (small integers, indeed for illustration while saving space) random samples from three different populations. With the null hypothesis: H 0 : µ1 = µ2 = µ3, and the alternative: H a : at least two of the means are not equal. At the significance level α = 0.05, the critical value from F-table is F 0.05, 2, 12 = 3.89. Sum Mean Sample P1 2 3 1 3 1 10 Sample P2 3 4 3 5 0 15 Sample P3 5 5 5 3 2 20 Demonstrate that, SST=SSB+SSW. That is, the sum of squares total (SST) equals sum of squares between (SSB) the groups plus sum of squares within (SSW) the groups. Computation of sample SST: With the grand mean = 3, first, start with taking the difference between each observation and the grand mean, and then square it for each data point. Sum Sample P1 1 0 4 0 4 9 Sample P2 0 1 0 4 9 14 Sample P3 4 4 4 0 1 13 Therefore SST = 36 with d.f = (n-1) = 15-1 = 14 Computation of sample SSB: Second, let all the data in each sample have the same value as the mean in that sample. This removes any variation WITHIN. Compute SS differences from the grand mean. Sum Sample P1 1 1 1 1 1 5 Sample P2 0 0 0 0 0 0 Sample P3 1 1 1 1 1 5 Therefore SSB = 10, with d.f = (m-1)= 3-1 = 2 for m=3 groups. Computation of sample SSW: Third, compute the SS difference within each sample using their own sample means. This provides SS deviation WITHIN all samples.
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