Researchers investigated the prevalence of personality disorders among adolescents who were psychiatric inpatients and had one of three diagnoses: (1) those with major depression, (2) those with substance abuse, and (3) those with both major depression and substance abuse. The mean number of disorders was as follows: major depression M = 1.1, substance abuse M = 0.6, those with both conditions M = 2.1. The researchers reported, "The three study groups differed in the average number of diagnosed personality disorders, F(2,123) = 10.31, p <0.0001." Explain this result to someone who is familiar with hypothesis testing with known populations but is unfamiliar with the t test or the analysis of variance. sum The null hypothesis is that the three groups are from populations of number of personality disorders with equal means. If this null hypothesis is true, the variance of these populations can be estimated in two ways. (i) Estimate from the variation within each of the three groups and then them. (ii) Estimate from the variation between the three means, multiplying by the size of each sample group. Since the estimates should be approximately equal, the ratio of the two estimates should be approximately 1. If the null hypothesis is false, the estimate (ii) is greater than the estimate (i) and the ratio, F, of (i) to (ii) is greater than 1. The fact that F = 10.31 is much larger than 1 and p <0.0001 indicates that the null hypothesis can be rejected.