P-Value

A -value is the probability, under a specified null hypothesis and its statistical model, of obtaining a test statistic at least as extreme as the observed value. The meaning of extreme is fixed by the test and its alternative hypothesis (Orloff and Bloom 2022, pp. 1-4). For a random variable representing the test statistic, with observed value , an upper-tailed test uses

(1)

where denotes probability calculated under the null hypothesis. A lower-tailed test uses

(2)

For a symmetric null distribution centered at 0, a conventional two-sided test uses

(3)

Thus a -value need not be an upper-tail probability. It does not give the probability that the null hypothesis itself is true (Wasserstein and Lazar 2016, p. 131).