This exercise motivates the formula for the between-subjects estimate of the variance in one-way ANOVA. Suppose each population mean equals μ (that is, H0 is true) and each sample size equals n. Then the sampling distribution of each i has mean μ and variance σ2/n, and the sample mean of the i values is the overall sample mean, {}.
a. Treating the g sample means as g observations having sample mean , explain why Σ (i - )2 / (g - 1) estimates the variance σ2/n of the distribution of the {i} values.
b. Using part a, explain why the between-groups estimate Σ n (i - )2 / (g - 1) estimates σ2. (For the unequal sample size case, the formula replaces n by ni.)