Statistics is Amazing

Here’s the Neyman-Pearson Lemma:  Suppose the likelihood ratio test with critical value has significance level  and power . Then any test with significance level or smaller has power less than .

So suppose we sample from some population with some distribution and have some hypothesis about what the parameters are for that distribution.  Then we want to devise some test to figure out when to accept our hypothesis and when to reject.  By controlling for type I error, we’ve found THE test that has the least probability of committing type II error.

Thus, in figuring out the distribution for some random phenomenon, we’ve devised a rather simple test and found that it has the least probability of accepting some hypothesis when it’s false for some specified probability of rejecting that hypothesis when it’s true!

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