In a two-proportion z-test, when testing the equality of two proportions, you typically use a pooled estimate of p under H0. Which statement is correct?

In a two-proportion z-test, when testing the equality of two proportions, you typically use a pooled estimate of p under H0. Which statement is correct?

• A. You pool the two sample proportions to estimate the common p
• B. You pool only the larger sample proportion
• C. You never pool
• D. You always pool for any sample size

Answer: (A)

When testing whether two proportions are equal, the null hypothesis asserts that the true proportions are the same. Under that assumption, you combine the data from both samples to estimate that common proportion, giving a single pooled estimate. This pooled p_hat is computed as (x1 + x2) / (n1 + n2), where x's are the number of successes and n's are the sample sizes. The test statistic then uses this pooled estimate to calculate the standard error: z = (p1_hat − p2_hat) / sqrt(p_hat_pooled × (1 − p_hat_pooled) × (1/n1 + 1/n2)). Using the pooled estimate reflects the null hypothesis of equality and leverages information from both samples to produce a more accurate standard error under that null. If you didn’t pool, you’d be assuming different proportions under the null, which isn’t what’s being tested. Note that with very small samples or extreme proportions, the normal approximation may be unreliable, and other methods may be preferred.

When testing whether two proportions are equal, the null hypothesis asserts that the true proportions are the same. Under that assumption, you combine the data from both samples to estimate that common proportion, giving a single pooled estimate. This pooled p_hat is computed as (x1 + x2) / (n1 + n2), where x's are the number of successes and n's are the sample sizes. The test statistic then uses this pooled estimate to calculate the standard error: z = (p1_hat − p2_hat) / sqrt(p_hat_pooled × (1 − p_hat_pooled) × (1/n1 + 1/n2)). Using the pooled estimate reflects the null hypothesis of equality and leverages information from both samples to produce a more accurate standard error under that null. If you didn’t pool, you’d be assuming different proportions under the null, which isn’t what’s being tested. Note that with very small samples or extreme proportions, the normal approximation may be unreliable, and other methods may be preferred.