Question: Consider the one-way ANOVA example with equal numbers of observations per group. Assuming that each observation has variance \(\sigma^{2}\), show the following:

1. If either the null or the alternative is true, \(E\left(M S_{\text {residual }}\right)=\sigma^{2}\)
2. If the null hypothesis is true (no differences between groups)

Hint: Use the fact that \(E X^{2}=\operatorname{var}(X)+(E X)^{2}\). Also show that \(\bar{y} . .=\frac{1}{J} \sum_{j} \bar{y}_{j}\)

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