Let $X_i \sim N(\mu_i, \sigma^2)$ for $i = 1, \dots, 4$. We want to test $H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4$ versus not $H_0$. We have three observations for each $i$. $X_1: 240, 221, 265, X_2: 286, 256, 272, X_3: 259, 245, 232$ and $X_4: 239, 215, 223$.
(a) Give a critical region for $\alpha = 0.05$.
(b) Construct an ANOVA table. What is your decision?
(c) What is your decision if $\alpha = 0.025$?
(d) What is the approximate $p$-value of the test?