4.5 Interpreting p-Values

What a p-value means

The p-value is the probability of getting a test statistic as extreme or more extreme than the observed result, assuming the null hypothesis (H₀) is true. It measures how surprising the sample data would be if H₀ were actually correct — it does NOT tell you the probability that H₀ is true.

Comparing p-value to alpha

Compare the p-value to the significance level (α), often 0.05. If p-value ≤ α, the result is statistically significant, so we reject H₀ in favor of the alternative hypothesis (Hₐ). If p-value > α, we fail to reject H₀ — there isn't enough evidence to support Hₐ.

Common misinterpretations to avoid

A small p-value does NOT prove Hₐ is true or measure effect size/importance — it only measures evidence against H₀. Never say we "accept H₀"; we only fail to reject it. Also, p-value is NOT the probability the null hypothesis is true, nor the probability of making an error.

Key terms

p-value (definition)

The probability of getting a test statistic as extreme or more extreme than observed, assuming H₀ is true.

Decision rule for p-value

If p-value ≤ α, reject H₀; if p-value > α, fail to reject H₀.

Misinterpretation to avoid

A p-value is NOT the probability that H₀ (or Hₐ) is true — it's a probability about the data given H₀.

"Fail to reject H₀" vs "Accept H₀"

We only fail to reject H₀ (insufficient evidence); we never say we 'accept' H₀ as proven true.

Small p-value meaning

Indicates the observed data would be unlikely if H₀ were true, providing evidence against H₀.

Try an AP-style question

t = −2 with df = 35 in a left-tailed test. What is the approximate p-value?

1. 0.05
2. 0.027
3. 0.0027
4. 0.973