Editing P-value (section)

=== 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 <math>\alpha</math> is obtained by rejecting the null-hypothesis if the ''p''-value is less than or equal to <math>\alpha</math>.<ref name="Bhattacharya2002">{{cite journal | vauthors = Bhattacharya B, Habtzghi D |s2cid = 33812107 |year = 2002 |title = Median of the p value under the alternative hypothesis |journal = The American Statistician |volume = 56 |issue = 3 |pages = 202–6 |doi = 10.1198/000313002146 }}</ref><ref name="Hung1997">{{cite journal | vauthors = Hung HM, O'Neill RT, Bauer P, Köhne K | title = The behavior of the P-value when the alternative hypothesis is true | journal = Biometrics | volume = 53 | issue = 1 | pages = 11–22 | date = March 1997 | pmid = 9147587 | doi = 10.2307/2533093 | type = Submitted manuscript | jstor = 2533093 | url = https://zenodo.org/record/1235121 }}</ref>

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 (<math>H_0: \mu \leq 0</math>, variance known), the null hypothesis does not specify the exact probability distribution of the appropriate test statistic. In this example that would be the [[Standard score|''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: <math>\alpha = 0.05</math> means one only rejects the null hypothesis if the ''p''-value is less than or equal to <math>0.05</math>, and the hypothesis test will indeed have a ''maximum'' type-1 error rate of <math>0.05</math>.