Mathfolis

Confidence Interval for the Difference of Two Proportions

Unit 6 · Inference for Categorical Data: Proportions

What AP Stats asks here

A two-proportion CI estimates the difference p1−p2p_1 - p_2 using two independent samples. The SE is unpooled — it does not assume p1=p2p_1 = p_2. If the CI excludes 0, there is evidence of a difference; if it contains 0, there is not. The common mistake: declaring 'no difference' because two one-sample CIs overlap. That heuristic is too conservative — only the difference CI is the correct decision tool.

Confidence interval

Estimate ± ME
(p^1−p^2)±z∗p^1(1−p^1)n1+p^2(1−p^2)n2(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}

Conditions

Independent random samples
required\text{required}
Large counts (each)
nip^i≥10 and ni(1−p^i)≥10n_i \hat{p}_i \ge 10 \text{ and } n_i (1-\hat{p}_i) \ge 10
10% (each)
ni<0.10Nin_i < 0.10 N_i
AP Tip: The smaller sample is the SE bottleneck. To shrink the difference CI, grow the smaller group first — doubling the larger group barely moves the needle.
Caution: Two one-sample CIs that overlap do not prove no difference. The two-sample CI uses the correct SE and is the right decision tool.
Type 1

Construct and interpret

Use the unpooled SE — never assume p1=p2p_1 = p_2 in a CI. Then state the interval as a range for the difference.

Example 1
City A: 66 of 300 random adults smoke. City B: 40 of 250 random adults smoke. Which is the 95% CI for pA−pBp_A - p_B? (A) (0.00,0.12)(0.00, 0.12) (B) (−0.005,0.125)(-0.005, 0.125) (C) (−0.10,0.22)(-0.10, 0.22) (D) (0.05,0.07)(0.05, 0.07)

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Type 2

Decision based on whether 0 is in the CI

Excluding 0 ⇒ evidence of a difference. Containing 0 ⇒ no convincing evidence of a difference.

Example 2
Which CI for p1−p2p_1 - p_2 provides evidence of a difference? (A) (−0.04,0.06)(-0.04, 0.06) (B) (−0.005,0.125)(-0.005, 0.125) (C) (0.02,0.15)(0.02, 0.15) (D) (0.00,0.12)(0.00, 0.12)

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Type 3

Use the difference CI, not overlap of two one-sample CIs

Overlap of one-sample CIs is a conservative check; only the two-sample CI is the correct tool for deciding whether the two proportions differ.

Example 3
City A 95% CI is (0.173,0.267)(0.173, 0.267) and City B 95% CI is (0.115,0.206)(0.115, 0.206) — they overlap in (0.173,0.206)(0.173, 0.206). The two-proportion CI for pA−pBp_A - p_B is (−0.005,0.125)(-0.005, 0.125). Which interpretation is best? (A) The one-sample CIs overlap, so the two cities must have the same smoking rate. (B) The two-sample CI is the proper decision tool. It contains 0, so we cannot conclude the rates differ — but overlap of the one-sample CIs alone does not prove equality. (C) The two-sample CI is wrong because the one-sample CIs already settle the question. (D) Without a hypothesis test, no conclusion is possible.

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