This lemma gives an optimum decision procedure for hypothesis testing. Suppose you have 2 hypotheses, H_{0} and H_{1}. You wish to know which, in fact, holds, so you perform an experiment and observe the results O (for our purposes, O can be the result of a series of experiments). The Nayman-Pearson lemma tells you that you should check if this holds (the parameter t will be explained later):

P(H_{1} | O) / P(H_{0} | O) > t

If it holds, you should decide the hypothesis H_{1} holds; otherwise H_{0} (sometimes called the null hypothesis) holds. Of course, larger values of t make it harder for you to decide the H_{1} holds. The Nayman-Pearson lemma says that for every (positive) value of t, this procedure is optimal in the following sense:

Thus, we see that the value t controls the ratio of false negatives to false positives. Different applications will require different ratios; we choose it to give the desired ratio.