The p-value is the probability, calculated under H_0, of obtaining a test statistic at least as extreme in the direction specified by H_1 as the observed statistic. **Alternative** | **p-value** H_1:mu< _0 | p=P(Z <= z_ obs) H_1:mu> _0 | p=P(Z >= z_ obs) H_1:mu!= _0 | p=2P(Z >= |z_ obs|) Reject H_0 when p <= alpha. Do not reject H_0 when p>alpha. Equality belongs to the rejection boundary under this convention. Deducing inequalities - To find a range of significance levels, compare alpha directly with the calculated p-value. - To find a range of a null parameter such as _0, write the relevant signed critical-region inequality and solve it algebraically. - Keep the test statistic, tail direction and parameter inequality consistent; multiplying by a negative quantity reverses an inequality. Always use |z_ obs| in a symmetric two-tailed p-value. The formula 1-P(-z_ obs<Z<z_ obs) fails when z_ obs<0.

The p-value is the probability, calculated under \(H_0\), of obtaining a test statistic at least as extreme in the direction specified by \(H_1\) as the observed statistic.

Reject \(H_0\) when \(p\leq\alpha\). Do not reject \(H_0\) when \(p>\alpha\). Equality belongs to the rejection boundary under this convention.

Decision boundary

To find a range of significance levels, compare \(\alpha\) directly with the calculated p-value.

To find a range of a null parameter such as \(\mu_0\), write the relevant signed critical-region inequality and solve it algebraically.

Keep the test statistic, tail direction and parameter inequality consistent; multiplying by a negative quantity reverses an inequality.

Always use \(|z_{\rm obs}|\) in a symmetric two-tailed p-value. The formula \(1-P(-z_{\rm obs}<Z<z_{\rm obs})\) fails when \(z_{\rm obs}<0\).

Alternative	p-value
\(H_1:\mu<\mu_0\)	\(p=P(Z\leq z_{\rm obs})\)
\(H_1:\mu>\mu_0\)	\(p=P(Z\geq z_{\rm obs})\)
\(H_1:\mu\ne\mu_0\)	\(p=2P(Z\geq \|z_{\rm obs}\|)\)

p-Values, Decisions and Parameter Inequalities

p-Values, Decisions and Parameter Inequalities