Unit 6: Inference for Categorical Data: Proportions

Significance Test for a Proportion

Conducting a one-sample significance test for a population proportion.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Confidence Interval for a Proportion

Testing Whether a Claimed Proportion Holds Up

A cereal company claims 20% of boxes contain a prize. A consumer group suspects the true rate is lower and checks a random sample of 150 boxes, finding 21 with a prize. Before reading on: how far below 20% would a sample result have to fall before you'd stop believing it's just ordinary sampling variability?

Definition — Significance Test for a Proportion

A significance test starts from a null hypothesis H₀: p = p₀ (the claimed value) and an alternative Ha, then asks how surprising the observed p̂ would be if H₀ were actually true. The test statistic is z = (p̂ − p₀)/√(p₀(1−p₀)/n) — using the hypothesized p₀, not p̂, in the standard error, since the test is evaluated under the assumption that H₀ is true.

z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}

Worked Example — Conducting a One-Sided Significance Test

H₀: p = 0.20. Ha: p < 0.20. p̂ = 21/150 = 0.14. Conditions: random (given), 10% condition (large box population), large counts using p₀: 150(0.20)=30≥10 and 150(0.80)=120≥10 — satisfied. SE = √(0.20×0.80/150) ≈ 0.0327. z = (0.14−0.20)/0.0327 ≈ −1.84. The p-value, P(Z ≤ −1.84), is about 0.033.

The correct interpretation: 'Assuming the true proportion of boxes with a prize really is 20%, the probability of getting a sample proportion as low as 0.14 (or lower) in a random sample of 150 boxes, just by chance, is about 0.033.' It does NOT mean 'there's a 3.3% probability that H₀ is true' — the p-value is computed by assuming H₀ is true from the start; it can't simultaneously be a probability that the assumption itself is correct. Since 0.033 < α = 0.05, reject H₀: there is convincing evidence the true proportion is less than 20%.

Tip

'Fail to reject H₀' is never the same claim as 'H₀ is proven true' — it only means this particular sample didn't provide strong enough evidence against it, at the chosen significance level.

Common Mistakes

Interpreting the p-value (0.033) as the probability that the null hypothesis is true.
The p-value is computed under the assumption H₀ is true — it measures how surprising the data would be in that world, not the probability that world is the real one.
Using a two-sided p-value when the consumer group's suspicion was specifically that the proportion is lower, not just different.
The alternative hypothesis should match the actual question being asked — 'is it lower' calls for a one-sided test (Ha: p < 0.20), which uses only the lower tail of the sampling distribution.

Key Takeaways

A significance test's standard error uses the hypothesized value p₀, not the sample's own p̂, since the test evaluates surprise under the assumption H₀ is true.
A p-value measures how likely a result this extreme (or more) is, assuming H₀ is true — it is never the probability that H₀ itself is true.
Failing to reject H₀ means insufficient evidence against it, not proof that H₀ is correct.

Summary

This closes Unit 6: inference for proportions estimates and tests categorical population parameters. Units 7 turns to quantitative data, where the population standard deviation is typically unknown — requiring a new distribution.

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