Significance Test for the Difference of Two Proportions

Unit 6 · Inference for Categorical Data: Proportions

What AP Stats asks here

The two-proportion z-test asks whether $p_1$ and $p_2$ are equal. Because $H_0$ asserts equality, the test pools the data into a common $\hat{p}_c$ for the SE. The CI, which never assumes equality, uses an unpooled SE. The trademark mistake is using one when you should use the other.

Test statistic

Pooled proportion

\hat{p}_c = \dfrac{x_1 + x_2}{n_1 + n_2}

z-statistic

z = \dfrac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}_c (1 - \hat{p}_c)(1/n_1 + 1/n_2)}}

Pool or not?

Test (

H_0: p_1 = p_2

)

\text{pool}

CI for

p_1 - p_2

\text{do not pool}

AP Tip: Mnemonic: Test of equality ⇒ pool. Estimate of difference ⇒ don't pool. The hypothesis test says 'imagine they're equal, see how unusual the data are'; the CI says 'estimate the difference'.

Caution: Test SE and CI SE differ slightly. A borderline test result and a CI almost touching 0 can occasionally disagree by a tiny amount — that mismatch is expected.

Type 1

Pooled SE (test) vs unpooled SE (CI)

Pool when the hypothesis is equality; otherwise use the unpooled SE.

Example 1

For each task, decide whether to pool. (i) Build a 95% CI for

p_1 - p_2

. (ii) Test

H_0: p_1 = p_2

\alpha = 0.05

. (iii) Estimate by how much smoking has declined from 1990 to 2025. Which matches? (A) (i) pool; (ii) pool; (iii) pool (B) (i) unpooled; (ii) pooled; (iii) unpooled (C) (i) pooled; (ii) unpooled; (iii) pooled (D) Always unpooled.

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Type 2

Compute z and p-value

With $\hat{p}_c$ in hand, compute z and translate to a one- or two-sided p-value.

Example 2

Drug A: 84 of 200 had side effects. Drug B: 60 of 200 had side effects. For a two-sided test at

\alpha = 0.05

: (A)

z = 2.50

, p-value ≈ 0.012, reject — evidence that side-effect rates differ. (B)

z = 1.50

, p-value ≈ 0.13, fail to reject. (C)

z = 2.50

, p-value ≈ 0.006, fail to reject. (D)

z = 0.50

, p-value ≈ 0.62, fail to reject.

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Type 3

CI and two-sided test agreement

A two-sided test rejecting $H_0$ at level $\alpha$ corresponds to the $(1-\alpha)$ CI excluding 0 (modulo the small pooled-vs-unpooled SE mismatch).

Example 3

Drug study: test gives

z = 2.50

, p-value 0.012, reject

H_0

\alpha = 0.05

. Unpooled 95% CI for

p_A - p_B

(0.027, 0.213)

. Which interpretation is correct? (A) The CI contradicts the test — fail to reject. (B) The CI excludes 0, matching the test's rejection. The pooled test SE and the unpooled CI SE differ slightly, but the conclusions agree: there is evidence that the side-effect rates differ. (C) The CI proves the test is wrong. (D) The two procedures are unrelated.

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