Let $X_i\sim Binomial(n,p)$. Consider the following hypotheses \begin{equation} (a): H_0: p = p_0 \quad \mbox{versus}\quad H_1: p = p_1 \end{equation} \begin{equation} (b): H_0: p = p_1 \quad \mbox{versus}\quad H_1: p = p_0 \end{equation}
If $(a)$ is rejected, what can be said about $(b)$? Is there any relation between these to hypotheses in terms of critical region?
It depends on the level of significance of the test. If the level of significance $\alpha_0=P(\text{reject $H_0$}|H_0)$ is small so that the rejection region has no overlap, then rejection of the hypothesis in (a) will necessarily mean that the hypothesis in (b) is not rejected, which is good. In particular, these cases are possible:
Neither hypothesis is rejected [contradictory conclusions]
(a) is rejected and (b) is not rejected [equivalent conclusions]
(b) is rejected and (a) is not rejected [equivalent conclusions]
However, if the level of significance is high enough that there is overlap in the rejection region, then it is possible for both hypotheses (a) and (b) to be rejected, if say the observed data point is between the two vertical lines.
