structural:expansion
Distribution for composite hypothesis

In parametric hypothesis testing problems, a simple or point hypothesis refers to a hypothesis where the parameter's value is assumed to be a single number. In contrast, in a composite hypothesis the parameter's value is given by a set of numbers. When the null-hypothesis is composite (or the distribution of the statistic is discrete), then when the null-hypothesis is true the probability of obtaining a p -value less than or equal to any number between 0 and 1 is still less than or equal to that number. In other words, it remains the case that very small values are relatively unlikely if the null-hypothesis is true, and that a significance test at level α {\displaystyle \alpha } is obtained by rejecting the null-hypothesis if the p -value is less than or equal to α {\displaystyle \alpha }.

For example, when testing the null hypothesis that a distribution is normal with a mean less than or equal to zero against the alternative that the mean is greater than zero (H 0: μ ≤ 0 {\displaystyle H_{0}:\mu \leq 0}, variance known), the null hypothesis does not specify the exact probability distribution of the appropriate test statistic. In this example that would be the Z -statistic belonging to the one-sided one-sample Z -test. For each possible value of the theoretical mean, the Z -test statistic has a different probability distribution. In these circumstances the p -value is defined by taking the least favorable null-hypothesis case, which is typically on the border between null and alternative. This definition ensures the complementarity of p-values and alpha-levels: α = 0.05 {\displaystyle \alpha =0.05} means one only rejects the null hypothesis if the p -value is less than or equal to 0.05 {\displaystyle 0.05}, and the hypothesis test will indeed have a maximum type-1 error rate of 0.05 {\displaystyle 0.05}.