Significance Tests for a Proportion
A significance test for a proportion weighs sample evidence against a claim about a population proportion, asking whether the result you saw would be surprising if the claim were true.
Definition
A significance test (hypothesis test) for a proportion starts from a null hypothesis H₀: p = p₀ (a specific claimed value) and an alternative Hₐ (that p is less than, greater than, or simply different from p₀). You measure how far your sample proportion falls from p₀ in standard-error units, then find the probability of a result at least that extreme assuming H₀ is true — the p-value.
The formula & procedure
The test statistic is z = (p̂ − p₀) / √( p₀(1 − p₀) / n ). Crucially, the standard error uses the null value p₀, not p̂. Conditions: random sample, the 10% condition, and large counts checked with p₀ — n·p₀ ≥ 10 and n(1 − p₀) ≥ 10. Find the p-value from the Normal distribution in the direction of Hₐ, then compare it to the significance level α (often 0.05): if the p-value < α, reject H₀; otherwise fail to reject.
Worked example
A podcast claims 30% of listeners finish every episode. A skeptic randomly surveys 150 listeners and finds 36 who finish every episode, so p̂ = 36/150 = 0.24. Test H₀: p = 0.30 against Hₐ: p < 0.30. The standard error is √(0.30·0.70/150) = √0.0014 ≈ 0.0374, so z = (0.24 − 0.30)/0.0374 ≈ −1.60. The left-tail p-value for z = −1.60 is about 0.055. Since 0.055 > 0.05, we fail to reject H₀ — the sample runs a little low, but not strongly enough at the 5% level to overturn the 30% claim.
The classic mistake
Using p̂ instead of p₀ in the standard error of the test statistic — that mixes up the test formula with the confidence-interval formula. Two more: "accepting" the null hypothesis, when failing to reject H₀ only means there isn't enough evidence against it, and reading the p-value as "the probability H₀ is true," which it is not.
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