8.3 Two-Way Classification (Unbalanced): 8.3.1 Representation as a Regression Model, 8.3.2 Hypothesis Testing, 8.3.3 Procedures for Testing the Hypotheses, 8.3.4 Confidence Intervals

Prove that $H_{A B}, H_{A B 1}, H_{A B 2}$, and $H_{A B 3}$ are all equivalent to $\mu_{i j} = \mu +\alpha_{i}+\beta_{j}$.

Using the syratetric constraints, exprese the parameters $\mu , \alpha_{i}$, etc. in terms of the $\mu_{i j}$.

Obtain an $F$-test statistic for testing $H: \mu_{i j} = \mu$ in a two-way classification. Find the expected value of the numerator sum of squares of your test statistics.