Chapter 13.5, Problem 18E

Refer to Exercise 13.17 and consider ${\bar{Y}}_{i•}-{\bar{Y}}_{i\prime •}$ for i ≠ i′.

a Show that $E({\bar{Y}}_{i•}-{\bar{Y}}_{i\prime •})={\mu }_i-{\mu }_{i\prime }={\tau }_i-{\tau }_{i\prime }$. This result implies that ${\bar{Y}}_{i•}-{\bar{Y}}_{i\prime •}$ is an unbiased estimator of the difference in the effects of treatments i and i′.

b Derive $V({\bar{Y}}_{i•}-{\bar{Y}}_{i\prime •})$.

13.17 Let ${\bar{Y}}_{i•}$ denote the average of all of the responses to treatment i. Use the model for the one-way layout to derive $E({\bar{Y}}_{i•})$ and $V({\bar{Y}}_{i•}).$