# Most powerful test

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

You're making the assumption that the two regions (C and D) relate to the same function of the observations. They don't (indeed, they can't).

Instead imagine the entire sample space (it may help to envisage a discrete distribution on very few observations -- for example, imagine a set of 6 observations that are either 0 or 1. Imagine that you want $\alpha = 3/64 \approx 0.047$ (3/64 being achievable in this case). Then you label can any three possible samples as "C" and any other three possible samples as "D".

Here's three possible samples:

  i: {0,1,0,1,0,1}
ii: {0,1,1,0,0,1}
iii: {0,1,1,1,0,0}


here's three more:

 iv: {0,0,1,1,0,1}
v: {0,0,1,0,1,1}
vi: {0,0,0,1,1,1}


So the first 3 might be "C" and the next three might be "D" (except more generally, C and D may have possible samples in common).

So anyway, more generally, you label a subset of those possible sample points with "C" and another subset (possibly overlapping the first) with "D".

If the sample being tested lies in the region labelled "C", it would result in rejection by that rejection rule.

Some possible rejection regions (labels on the sample space) will be better at rejecting a given alternative than others. ... (Now try rereading the page you were confused by.)