I don't understand what is meant by a most powerful test. I was reading the article here and the definition of most powerful test is given as,
Definition. Consider the test of the simple null hypothesis $H_0: θ = θ_0$ against the simple alternative hypothesis $H_A: θ = θ_a.$ Let C and D be critical regions of size α, that is, let:
$α=P(C;θ_0)$ and $α=P(D;θ_0)$
Then, C is a best critical region of size α if the power of the test at $θ = θ_a$ is the largest among all possible hypothesis tests. More formally, C is the best critical region of size α if, for every other critical region D of size α, we have:
$P(C;θ_α)≥P(D;θ_α)$ that is, C is the best critical region of size α if the power of C is at least as great as the power of every other critical region D of size α. We say that C is the most powerful size α test.
I don't understand how two can there be two critical regions of size $\alpha$ as C and D. For a Z test how I find critical region for say $Ha:\theta>\theta_0$ with 0.05 significant level is, I take the Z value that gives a probability of 0.05, that is Z=1.96 and the region right to 1.96 is the critical region . So how come there be two critical regions C and D.
Can someone provide me with an example