12.44.* This exercise motivates the formula for the between-groups variance estimate in one-way ANOVA.

Suppose the sample sizes all equal n and the population means all equal μ. The sampling distribution of each ¯yi then has mean μ and variance σ2/n. The sample mean of the ¯yi values is ¯y.

(a) Treating ¯y1, ¯y2, . . . , ¯yg as g observations having sample mean ¯y, explain why



( ¯yi− ¯y)2/(g−1) estimates the variance

σ2/n of the sampling distribution of the ¯yi-values.

(b) Using (a), explain why



n( ¯yi − ¯y)2/(g−1) estimates

σ2. For the unequal sample size case, replacing n by ni yields the between-groups estimate.