4.6 Concluding a Test for a Population Mean

Making the decision

Compare the p-value to the significance level α. If p-value ≤ α, reject H₀ — the data provide convincing statistical evidence for Hₐ. If p-value > α, fail to reject H₀ — there is not convincing evidence for Hₐ. You never 'accept' H₀; you simply lack enough evidence to reject it.

Stating conclusions in context

A conclusion must reference the context of the problem, the p-value, and α. Example: 'Since the p-value (0.02) < α (0.05), we reject H₀. There is convincing evidence that the true mean [context] differs from [claimed value].' Avoid saying the alternative is 'true' or 'proven.'

Errors and their meaning

A Type I error occurs if we reject a true H₀ (false positive); its probability equals α. A Type II error occurs if we fail to reject a false H₀ (false negative). Conclusions should acknowledge that some risk of error always remains — statistical tests give evidence, not certainty.

Key terms

Decision rule for hypothesis tests

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

Type I error

Rejecting a true null hypothesis; probability = α.

Type II error

Failing to reject a false null hypothesis.

Proper conclusion wording

State decision, p-value vs. α comparison, and context — e.g., 'Since p-value < α, we reject H₀; there is convincing evidence that...'

Why we never 'accept' H₀

Failing to reject H₀ only means insufficient evidence against it, not proof that it's true.

Try an AP-style question

α = 0.05 and p = 0.027 for H₀: μ = 12 vs. Hₐ: μ < 12. What is the conclusion?

1. Fail to reject H₀; the true mean battery life is 12 hours
2. Accept Hₐ; the true mean battery life is exactly 11.5 hours
3. Reject H₀; convincing evidence the true mean is less than 12 hours
4. Reject H₀; convincing evidence the sample mean is less than 12