For the model of a single factor experiment:
$y_{ij}= \mu + \alpha_i + e_{ij}$, $(1 \leq i \leq a, 1 \leq j \leq n_i)$,
where a = the number of treatments, $n_i$ = the number of experimental units assigned to each treatment, $\alpha_i$ = the deviations of the treatment means $\mu_i$.
The four main assumptions of the model are:
1) the errors $e_{ij}$ are normally distributed;
2) the errors $e_{ij}$ are homoscedastic;
3) the errors $e_{ij}$ are independently distributed; and
4) $E(e_{ij}) = 0$ or equivalently $E(y_{ij}) = \mu_i$
I'm trying to prove $Corr(\hat{e}_{ij}, \hat{e}_{jk}) = \frac{-1}{n_i-1}$ for $ j \neq k$ but I am stuck at the second step.
$Corr(\hat{e}_{ij}, \hat{e}_{jk}) = \frac{Cov((y_{ij}-\bar{y}_i),(y_{ik}-\bar{y}_i))}{\sigma_{ij}\sigma_{ij}}$
I know the usual formula for corr(X,Y) is:
$\frac{E(X-\mu_x)(Y-\mu_y)}{\sigma_x \sigma_y}$
However, I'm not sure how to proceed.