Which expression gives the standard error used in a 95% confidence interval for a proportion p_hat with n = 100 and p_hat = 0.40?

• A. sqrt(p_hat(1-p_hat)/n)
• B. sqrt((1-p_hat)/n)
• C. sqrt(p_hat/n)
• D. sqrt(p_hat^2/n)

Answer: (A)

The standard error for a proportion in a confidence interval uses the variability of a binomial proportion, which is p(1−p)/n. Since we estimate p with p_hat, the standard error becomes sqrt(p_hat(1−p_hat)/n). This is why the correct form is sqrt(p_hat(1−p_hat)/n). Plugging in n = 100 and p_hat = 0.40 gives sqrt(0.40 × 0.60 / 100) = sqrt(0.24 / 100) = sqrt(0.0024) ≈ 0.049. The other expressions fail because they omit either the p_hat or (1−p_hat) factor, or they square only p_hat, which doesn't reflect the true variability of a proportion.