The decision to accept or reject a hypothesis is associated with the so-called significance and power of the test. The significance is the probability, according to the null hypothesis \(h\), of obtaining data that leads us to falsely reject this hypothesis \(h\):  \[ \text{Significance}_{F} = \alpha = P_{h}(R_{r}) = \sum_{s \in S} F(s) P_{h}(s) d s , \]  The probability \(\alpha\) is alternatively called the type-I error, and it is often denoted as the significance or the p-value. The power is the probability, according to the alternative hypothesis \(h'\), of obtaining data that leads us to correctly reject the null hypothesis \(h\):  \[ \text{Power}_{F} = 1 - \beta = P_{h'}(F_{1}) = \sum_{s \in S} F(s) P_{h'}(s) d s.  \]  The probability \(\beta\) is called the type-II error of falsely accepting the null hypothesis. An optimal test is one that minimizes both the errors \(\alpha\) and \(\beta\). In their fundamental lemma, Neyman and Pearson proved that the decision has optimal significance and power for, and only for, likelihood-ratio test functions \(F\). That is, an optimal test depends only on a threshold for the ratio \(P_{h'}(s) / P_{h}(s)\).