6.10 Setting Up a Test for the Difference of Two Population Proportions

For a two-sample test of proportions you always start with the null that the populations are the same (difference = 0): H0: p1 = p2 or H0: p1 − p2 = 0. The alternative depends on your question: - Two-sided: Ha: p1 ≠ p2 (tests for any difference). - One-sided: Ha: p1 > p2 or Ha: p1 < p2 (tests a directional claim). On the AP exam you’ll use a two-sample z-test for proportions (when conditions are met). Under H0 you form the pooled proportion p̂c = (x1 + x2)/(n1 + n2) and use the pooled SE: sqrt[p̂c(1−p̂c)(1/n1 + 1/n2)] to compute the z statistic. Before testing check independence (random/independent samples, 10% rule if without replacement) and the success–failure condition using the pooled p̂c: n1p̂c, n1(1−p̂c), n2p̂c, n2(1−p̂c) ≥ 5 (or 10 per your class). For a short walkthrough and examples see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP). For extra practice, try problems on the Unit 6 page (https://library.fiveable.me/ap-statistics/unit-6) or the general practice bank (https://library.fiveable.me/practice/ap-statistics).

One-sample z-test for a proportion tests one population proportion against a hypothesized value: H0: p = p0. The test statistic is (p̂ − p0) / sqrt[p0(1−p0)/n] and you check independence (random sample, 10% condition) and success–failure using p0 (or p̂ for CI). Two-sample z-test compares two independent proportions: H0: p1 = p2 (or p1 − p2 = 0) and Ha can be one- or two-sided (p1 ≠ p2, p1 > p2, p1 < p2). Under H0 you use the pooled proportion p̂c = (x1 + x2)/(n1 + n2) to compute the pooled standard error: SE = sqrt[p̂c(1−p̂c)(1/n1 + 1/n2)]. Check independence (two independent random samples or experiment + 10% for each) and success–failure using the pooled counts: n1p̂c, n1(1−p̂c), n2p̂c, n2(1−p̂c) ≥ 5 (or 10). AP tip: the CED expects a two-sample z-test for comparing proportions (Topic 6.10). Review pooled/stat formula and conditions at the Fiveable study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and practice problems (https://library.fiveable.me/practice/ap-statistics).

Use the two-sample z-test for the difference of two population proportions when you have two independent random samples (or a randomized experiment) and you want to test H0: p1 = p2 (or p1 − p2 = 0). Under H0 you use the pooled proportion p̂c = (x1 + x2)/(n1 + n2) to compute the pooled standard error and z statistic. Before using the test check the CED conditions: independence (random samples, 10% rule when sampling without replacement) and approximate normality via the success–failure condition using the pooled p̂c—i.e., n1 p̂c, n1(1−p̂c), n2 p̂c, n2(1−p̂c) all ≥ about 5 (or 10 for stricter practice). When not to use it: if samples aren’t independent, counts are small (use Fisher’s exact or exact methods), or you’re doing one proportion (use one-sample z-test) or comparing many categories (chi-square). For CI for p1 − p2 you use the two-sample z-interval formula (unpooled SE). For more AP-aligned detail and examples see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and the Unit 6 overview (https://library.fiveable.me/ap-statistics/unit-6). For extra practice try the problems at (https://library.fiveable.me/practice/ap-statistics).

p̂c (the pooled proportion) is the combined estimate of the common proportion under H₀: p₁ = p₂. Compute it two equivalent ways: - p̂c = (n₁·p̂₁ + n₂·p̂₂) / (n₁ + n₂) - or, if you know counts of successes, p̂c = (X₁ + X₂) / (n₁ + n₂) You use p̂c when doing a two-sample z-test for p₁ − p₂ because under the null you assume the two populations have the same p, so you pool the data to estimate that common p. Then the pooled standard error is sqrt[ p̂c(1−p̂c) (1/n₁ + 1/n₂) ] and z = (p̂₁ − p̂₂) / (pooled SE). Don’t forget the AP conditions: samples independent, 10% rule if sampling without replacement, and the success–failure checks using p̂c (n₁p̂c, n₁(1−p̂c), n₂p̂c, n₂(1−p̂c) ≥ 5 or 10). See the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and Unit 6 overview (https://library.fiveable.me/ap-statistics/unit-6) for examples and practice (https://library.fiveable.me/practice/ap-statistics).

H₀: p₁ − p₂ = 0 means you’re claiming the two population proportions are equal—there’s no difference or effect between group 1 and group 2. In practice that means your test is set up so the center of the sampling distribution for p̂₁ − p̂₂ is 0 under the null. For AP Stats you’d use a two-sample z-test for proportions (CED VAR-6.I.1), compute the pooled proportion p̂c = (X₁+X₂)/(n₁+n₂) when assuming H₀, and use the pooled standard error √[p̂c(1−p̂c)(1/n₁ + 1/n₂)] to get the z statistic. Before you trust the z test, check independence (random, independent samples and 10% rule) and the success–failure condition using n₁p̂c, n₁(1−p̂c), n₂p̂c, n₂(1−p̂c) ≥ 5 or 10 (CED VAR-6.J). For a quick refresher on setup and pooled SE, see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP). If you want practice problems, Fiveable also has lots at (https://library.fiveable.me/practice/ap-statistics).

Step-by-step check for a two-sample z-test for p1 − p2 (assume H0: p1 = p2): 1. State H0 and Ha. For example H0: p1 − p2 = 0 (p1 = p2). Choose one- or two-sided Ha (p1 > p2, p1 < p2, or p1 ≠ p2). 2. Independence: confirm data come from two independent random samples or a randomized experiment. If sampling without replacement, check each sample size ≤ 10% of its population (n1 ≤ 0.10N1 and n2 ≤ 0.10N2). 3. Pooled proportion (use for the test): p̂c = (x1 + x2) / (n1 + n2), where x1 = n1·p̂1 and x2 = n2·p̂2. 4. Success–failure (normality): compute n1·p̂c, n1·(1−p̂c), n2·p̂c, n2·(1−p̂c). Each should be ≥ the rule-of-thumb (commonly 5, sometimes 10) so the sampling distribution of p̂1 − p̂2 is approximately normal. 5. If all checks pass, use z = (p̂1 − p̂2 − 0) / sqrt[p̂c(1−p̂c)(1/n1 + 1/n2)] to get p-value. If not, use a simulation/bootstrap or exact method. For AP alignment and examples, see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and Unit 6 overview (https://library.fiveable.me/ap-statistics/unit-6). Practice more problems at (https://library.fiveable.me/practice/ap-statistics).

You check n₁p̂c, n₁(1−p̂c), n₂p̂c, n₂(1−p̂c) ≥ 5 (or ≥10) because the two-sample z-test for p₁−p₂ relies on the sampling distribution being approximately normal. Under H₀: p₁ = p₂ you pool the samples to get p̂c = (n₁p̂₁ + n₂p̂₂)/(n₁ + n₂); pooling gives the best estimate of the common proportion and sets the correct standard error for the z statistic. If any of those counts are too small, the normal approximation is poor and the z-test (and its p-value) can be misleading. You also must check independence: samples should be random/independent and, when sampling without replacement, n ≤ 10% of the population (for each sample). These checks are required by the CED (VAR-6.J) for valid inference on the AP exam. For a quick refresher and examples, see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and try practice questions (https://library.fiveable.me/practice/ap-statistics).

Use a two-sample z-test for proportions when you want to test H0: p1 = p2 (or p1 − p2 = 0). Under H0 you pool the samples: - pooled proportion: p̂c = (x1 + x2) / (n1 + n2), where x1 = n1 p̂1 and x2 = n2 p̂2. - pooled standard error: SE = sqrt[ p̂c(1 − p̂c) (1/n1 + 1/n2) ]. - test statistic: z = (p̂1 − p̂2) / SE (since H0 assumes difference = 0). Also check conditions from the CED: independent random samples (and 10% rule when sampling without replacement) and success–failure using the pooled proportion: n1 p̂c, n1(1−p̂c), n2 p̂c, n2(1−p̂c) should be ≥ 5 (or ≥10 if your teacher wants the stricter rule) so the sampling distribution is approximately normal. For more AP-aligned review see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and the Unit 6 page (https://library.fiveable.me/ap-statistics/unit-6). For extra practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

Pick Ha based on the question you’re trying to answer—the alternative should match the claim or research question. - If you want to detect any difference (either group could be higher), use two-sided: Ha: p1 ≠ p2. This is common on the exam when they ask “different” or don’t indicate direction (CED VAR-6.H.3). - If you have a directional claim (you expect one proportion to be larger or smaller), use one-sided: Ha: p1 > p2 or Ha: p1 < p2. Use Ha: p1 > p2 when you’re testing whether population 1’s success rate is greater than population 2’s (CED VAR-6.H.3). - Don’t pick direction after seeing data—the direction must be decided before collecting/analyzing data. Also remember the null is H0: p1 = p2 (or p1 − p2 = 0) and, for a two-sample z-test, you’ll use the pooled proportion when H0 is assumed (CED VAR-6.H.1, VAR-6.I.1, VAR-6.J.1b). For a quick review on wording and examples, see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP). For more practice, Fiveable has lots of practice questions (https://library.fiveable.me/practice/ap-statistics).

“Independent random samples” means the two groups you compare were chosen separately so one sample doesn’t affect the other. For a two-sample z-test for p1 − p2 the CED requires either (a) two independent random samples from each population or (b) a randomized experiment that assigns units to two treatments. Practically that means you didn’t sample the same people twice (paired data) and you didn’t pick one sample based on the other. Why it matters: the standard error formula and pooled proportion (p̂c = (x1+x2)/(n1+n2)) assume independence. If samples aren’t independent you must use a paired method or a different test. Also check the 10% condition when sampling without replacement (n1 ≤ 10% N1 and n2 ≤ 10% N2) and the success–failure checks (n1 p̂c, n1(1−p̂c), n2 p̂c, n2(1−p̂c) ≥ 5 or 10) for approximate normality, per the CED. Want more practice or a quick refresher? See the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and thousands of practice problems (https://library.fiveable.me/practice/ap-statistics).

Check two things. First, independence: your data must come from two independent random samples or a randomized experiment, and if sampling without replacement each sample size should be ≤10% of its population (the 10% condition). Second, the sampling distribution must be approximately normal (success–failure condition). For a two-sample test you use the pooled proportion under H0: p̂c = (n1 p̂1 + n2 p̂2)/(n1 + n2). Then check that all four counts n1 p̂c, n1(1−p̂c), n2 p̂c, and n2(1−p̂c) are ≥ a cutoff (AP uses “typically 5 or 10”—many teachers use ≥5; some prefer ≥10 for extra caution). If they’re met, the two-sample z-test (pooled z) is OK; if not, don’t rely on the normal approx. For a quick refresher see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and other Unit 6 resources (https://library.fiveable.me/ap-statistics/unit-6). For extra practice try the AP question bank (https://library.fiveable.me/practice/ap-statistics).

The 10% rule is a quick check for independence when you sample without replacement: each sample size should be no more than 10% of its population (n₁ ≤ 0.10·N₁ and n₂ ≤ 0.10·N₂). Why? When you remove many units from a finite population, later draws aren’t independent of earlier ones. If each sample is ≤10% of the population, the dependence is negligible and you can treat observations as (approximately) independent—an AP requirement for inference (CED VAR-6.J.1.a.ii). On the AP exam you must verify independence before using the two-sample z-test for proportions. If you can’t meet the 10% rule, you either use sampling with replacement, adjust variance with a finite-population correction, or acknowledge the independence issue and avoid standard z-methods. For more on checks and the pooled-proportion z-test, see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and grab extra practice at (https://library.fiveable.me/practice/ap-statistics).

Pooled proportion is the single best estimate of the common population proportion you assume under the null H0: p1 = p2. You compute it as p̂c = (x1 + x2) / (n1 + n2) (equivalently (n1 p̂1 + n2 p̂2)/(n1 + n2)). We use p̂c when doing a two-sample z-test for p1 − p2 because the null says both populations share one p, so the standard error should be based on that common value. The pooled standard error is sqrt[p̂c(1−p̂c)(1/n1 + 1/n2)] and the z statistic is (p̂1 − p̂2)/SEpooled. Why this matters for AP: the CED requires you to assume p1 = p2 under H0 and to check the success–failure condition using the pooled count (n1 p̂c, n1(1−p̂c), n2 p̂c, n2(1−p̂c) ≥ 5 or 10). If those hold and samples are independent (and ≤10% rule if without replacement), the pooled z-test is appropriate. For a refresher, see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP). For more review and lots of practice problems, check the Unit 6 page (https://library.fiveable.me/ap-statistics/unit-6) and the practice bank (https://library.fiveable.me/practice/ap-statistics).

Use pooled when you’re doing a hypothesis test that assumes the two population proportions are equal under H₀ (H₀: p₁ = p₂ or p₁ − p₂ = 0). In that case you combine successes to get the pooled proportion p̂c = (x₁ + x₂)/(n₁ + n₂), use p̂c in the success–failure check (n₁p̂c, n₁(1−p̂c), n₂p̂c, n₂(1−p̂c) ≥ 5 or 10) and in the pooled SE: sqrt[p̂c(1−p̂c)(1/n₁ + 1/n₂)]. That’s the AP-approved approach for two-sample z-tests (CED VAR-6.I/J). Use the unpooled (separate p̂₁ and p̂₂) when you’re building a confidence interval for p₁ − p₂ or when the test does NOT assume p₁ = p₂—CIs always use the separate SE: sqrt[ p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂ ]. Also always check independence (random/independent samples and 10% condition). For a quick review see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and practice questions (https://library.fiveable.me/practice/ap-statistics).

Pick an order for your groups (call them p1 = proportion for group 1, p2 = proportion for group 2). For a one-sided test that group 1 has a larger proportion than group 2 you write: H0: p1 = p2 (equivalently p1 − p2 = 0) Ha: p1 > p2 (equivalently p1 − p2 > 0) This is a one-sided (right-tailed) two-sample z-test for proportions. Under H0 you’ll use the pooled proportion p̂c = (x1 + x2)/(n1 + n2) to compute the standard error and z statistic. Before testing, check independence (random/independent samples and 10% condition) and the success–failure condition using the pooled counts (n1 p̂c, n1(1−p̂c), n2 p̂c, n2(1−p̂c) usually ≥ 10). For AP-aligned review and examples see the Topic 6.10 study guide (https://library.fiveable.me/ap-statistics/unit-6/setting-up-test-for-difference-two-population-proportions/study-guide/sJt2F9NwsQ2gihYEHGjP) and extra practice (https://library.fiveable.me/practice/ap-statistics).