p-value explained
The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.
Rephrased: if H₀ were true, how often would you see data this surprising or more surprising? A small p-value means the data are unlikely under H₀, so you reject H₀. A large p-value means the data are consistent with H₀, so you fail to reject it.
A p-value is not: the probability H₀ is true, the probability you made an error, or the size of the effect.
Worked example
You test whether a process mean μ = 50. From a sample you compute a test statistic z = 2.3. Find the two-sided p-value.
At α = 0.05: p = 0.0214 < 0.05 ⇒ reject H₀. There is sufficient evidence the mean differs from 50.
One-sided vs two-sided
| Test | H₁ | p-value |
|---|---|---|
| Two-sided | μ ≠ μ₀ | 2 × P(Z > |z|) |
| Upper one-sided | μ > μ₀ | P(Z > z) |
| Lower one-sided | μ < μ₀ | P(Z < z) |
Practice problem
A one-sided upper test produces z = 1.7. What is the p-value? Should you reject at α = 0.05?
Show answer
p = P(Z > 1.7) = 1 − 0.9554 = 0.0446. Since 0.0446 < 0.05, reject H₀.
Common exam traps
- “p = 0.04 means 4% probability H₀ is true.” FALSE. The p-value conditions on H₀ being true. It does not give the probability H₀ is true.
- p > 0.05 does not prove H₀. Failing to reject is not the same as proving the null. You may simply have insufficient power (too small an n).
- p-values are sensitive to sample size. A trivially small effect can produce p < 0.001 with a very large sample. Always pair a p-value with an effect size or confidence interval.
- Pre-specify α before looking at the data. Choosing α after you see the p-value (p-hacking) inflates the type I error rate.
Try 5 free CQE practice questions on this exact topic. Every wrong answer comes with an explanation. No account needed.
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