Suppose we have g = 1,..., k iid samples $X_g \sim N(\ \mu_g, \sigma^2$). We know $\sigma^2$ and k sample means $m_g$ are given. We wanto to test
$H_o: \mu_g = a$, g = 1, ..., k
$H_a$: some $\mu_g ≠ a$
Propoesed testing procedure:
Tets statistics: $$\frac{\sum_{g=1}^k (m_g - \mathbf a)^2}{{\sigma^2/n}} \sim \chi^2(k)$$
and we use an upper-tail test.
Is this procedure correct? Note that this is different from the usual anova problem in two respects: 1. we are testing for the equality of means to specific value (a) not just equality 2. variance is known.
In fact in anova we implicitly have a = pooled mean and the variance is estimated.
Thanks