In a two-way factorial, we have defined  as the grand mean of the data,  + _I as the mean of the responses for the  th level of factor A  as the mean of the responses for the th level of factor B, and  as the mean of the th factor-level combination. Show that this implies our zero-sum restrictions on the estimated effects.

Suppose that we use the same idea, but instead of ordinary averages we use weighted averages with  as the weight for the th factor-level combination. Derive the new zero-sum restrictions for these weighted averages.