p-value

Definition taught in Hypothesis tests, part of Statistical inference

p-value

The caffeine test below. From Hypothesis tests.
Type Definition
Taught in Hypothesis tests
Part Statistical inference
Worked examples 2

The p-value of an observed statistic t is the probability, under H_0, that T is at least as extreme as t, in the direction of H_1. For a two-sided normal or t statistic it is twice the one-sided tail probability beyond |t|.

Examples

A gene in lake fish

Worked example from Hypothesis tests

A gene occurs in 15% of a fish population. In a sample of 100 fish from one lake, 10 carry it.

Question. Is there evidence that the lake population differs?

Solution. (1) Let X be the number of carriers among 100, X\sim\operatorname{Bin}(100,p). (2) H_0:p=0.15 against H_1:p\neq0.15; the alternative is the claim that the lake is different. (3) The statistic is X. (4) Under H_0, X\sim\operatorname{Bin}(100,0.15). (5) x=10. (6) The two-sided p-value is 2\mathbb{P}_{H_0}(X\le10)=2\times0.0994=0.199. The normal approximation \operatorname{N}(15,12.75) gives 2\Phi\{(10-15)/\sqrt{12.75}\}=2\Phi(-1.40)=0.161, and with the continuity correction, 2\Phi\{(10.5-15)/\sqrt{12.75}\}=0.208, close to the exact value. (7) The p-value is large, so H_0 is not rejected. (8) There is no evidence that the proportion of carriers in the lake differs from 15%.

Caffeine in cola

Worked example from Hypothesis tests

A claim is that regular cola contains 19 mg of caffeine per serving. For n=40 servings of another brand, \overline x=19.57 mg and s=1.40 mg.

Question. At level 0.01, is the mean caffeine content of this brand more than 19 mg?

Solution. Model the servings as a normal random sample. Test H_0:\mu=19 against H_1:\mu>19. The statistic is

t=\frac{19.57-19}{1.40/\sqrt{40}}=2.575,

against t_{39}. The critical value is t_{39;0.99}=2.43, so t is in the critical region, and the p-value is 0.007. Reject H_0: there is strong evidence that the mean exceeds 19 mg.

Statistical significance is not practical significance. The one-sided 99% lower confidence bound is 19.57-2.43\times1.40/\sqrt{40}=19.03, so the data are consistent with a mean barely above 19. The two-sided 99% interval, (18.97,\ 20.17), even includes 19, because it spends only half its error on each side. Whether a mean of 19.5 or 20 mg matters for health is a question the test cannot answer.

Figures

Left: a two-sided z test of size 0.05. Under H_0 the statistic is \operatorname{N}(0,1), the critical region is the two tails beyond \pm1.960, and an observed z=2.30 has p-value 0.021. Right: under the alternative Z\sim\operatorname{N}(2,1) the same cutoffs split the density into the power, the part beyond them, and the type II error, the part between. From Hypothesis tests.

Used in

See also

Null and alternative hypotheses · Critical region · Type I and type II errors · Neyman–Pearson approach · Power function · One-sample t test · Two-sample t test · F test for equal variances

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