3.8 Concluding a Test for a Population Proportion

Compare p-value to alpha

To conclude a significance test, compare the p-value to the significance level alpha (α). If p-value ≤ α, reject H₀ — the data provide convincing evidence for Hₐ. If p-value > α, fail to reject H₀ — there is not convincing evidence for Hₐ. Never say 'accept H₀' or 'prove.'

State conclusions in context

A proper conclusion links the decision to the context of the problem, not just symbols. Example: 'Because the p-value (0.02) is less than α = 0.05, we reject H₀. There is convincing evidence that the true proportion of... differs from p₀.' Always restate in terms of the population proportion, not the sample.

Errors don't disappear

Even with a correct conclusion, Type I or Type II errors remain possible since the decision is based on a sample. Rejecting H₀ when it's actually true is a Type I error (probability α); failing to reject H₀ when Hₐ is true is a Type II error. Conclusions are always probabilistic, not certain.

Key terms

Decision rule for hypothesis tests

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

Why we never 'accept H₀'

Failing to reject H₀ just means insufficient evidence against it — it doesn't prove H₀ is true.

Components of a full conclusion

State the decision (reject/fail to reject), justify with p-value vs. α, and interpret in context using the population proportion.

Type I error

Rejecting H₀ when it is actually true; occurs with probability α.

Type II error

Failing to reject H₀ when Hₐ is actually true.

Try an AP-style question

α = 0.05 and the p-value is 0.03. What is the conclusion?

1. Reject Hₐ in favor of H₀
2. Accept H₀ as true
3. Reject H₀; there's convincing evidence for Hₐ
4. Fail to reject H₀; there's no evidence for Hₐ