If you want to estimate the difference between two proportions with a confidence interval, which method should you use?

If you want to estimate the difference between two proportions with a confidence interval, which method should you use?

• A. 2-proportion z-interval
• B. 2-proportion z-test
• C. 1-proportion z-interval
• D. Chi-square interval

Answer: (A)

Estimating the difference between two proportions with a confidence interval relies on the idea that the difference in sample proportions is roughly normally distributed when both samples are large. This allows you to form a z-interval specifically for the difference p1 − p2. You take the observed difference p1_hat − p2_hat as the center, and you multiply the standard error by a z-score for your desired level of confidence. The standard error combines the variability from both samples: sqrt[ p1_hat(1 − p1_hat)/n1 + p2_hat(1 − p2_hat)/n2 ]. This reflects that each group contributes its own sampling variation, and the total uncertainty about the difference comes from both halves of the study. This approach is the best fit here because it targets the quantity of interest—the difference in proportions—and uses information from both groups to quantify uncertainty. The other options don’t directly provide a confidence interval for a difference in proportions: one-proportion intervals estimate a single proportion, a two-proportion z-test is for hypothesis testing rather than estimating a confidence interval, and a chi-square interval isn’t the standard method for this particular CI.

Estimating the difference between two proportions with a confidence interval relies on the idea that the difference in sample proportions is roughly normally distributed when both samples are large. This allows you to form a z-interval specifically for the difference p1 − p2. You take the observed difference p1_hat − p2_hat as the center, and you multiply the standard error by a z-score for your desired level of confidence.

The standard error combines the variability from both samples: sqrt[ p1_hat(1 − p1_hat)/n1 + p2_hat(1 − p2_hat)/n2 ]. This reflects that each group contributes its own sampling variation, and the total uncertainty about the difference comes from both halves of the study.

This approach is the best fit here because it targets the quantity of interest—the difference in proportions—and uses information from both groups to quantify uncertainty. The other options don’t directly provide a confidence interval for a difference in proportions: one-proportion intervals estimate a single proportion, a two-proportion z-test is for hypothesis testing rather than estimating a confidence interval, and a chi-square interval isn’t the standard method for this particular CI.