Upper Bound for Significance level (Equal means where the null would be rejected) I have been studying Power Analysis and ANOVA and wanted to know how I can find an upper bound for the significance level ($\alpha$), for which $H_0: \mu_1 = \mu_2 = ... = \mu_n$ would be rejected? Would appreciate an explanation or if there is any kind of computation/formula for this.
 A: $\alpha$ is the probability of a type I error which is where we reject the null despite it being true. Under the null our p-value $p$ is uniformly distributed on $[0,1]$, so the probability of rejecting a true null is $P(p \leq \alpha\mid H_0) = \alpha$. We reject at a significance level of $\alpha$ by comparing $p$ to $\alpha$.
An upper bound is easy since if we take $\alpha=1$ we'll always reject. By doing this we're saying that we're ok with always rejecting true nulls but at least we'll never fail to reject a false null. Our power is therefore $100\%$.
More interesting is the smallest $\alpha$ such that we'll reject, and this turns out to be exactly $p$. This is the whole appeal of the p-value: rather than simply getting a "reject or not" answer for some $\alpha$, the p-value tells us exactly at what level of significance we can reject and when we no longer have enough evidence to reject. If $p=0.54$, say, then $\alpha=0.54$ will let us reject but we'd fail to reject for any smaller $\alpha$, like the usual $\alpha=0.05$.
This applies to hypothesis testing in general and doesn't require any particular null like $H_0 : \mu_1 = \dots = \mu_n$. For that null we'd likely end up with an F-test which will lead us to our p-value and then we can see what range of $\alpha$ would let us reject.
