Construct a 95% confidence interval for a proportion p_hat = 0.40 with n = 100 (approx using standard formula).

Construct a 95% confidence interval for a proportion p_hat = 0.40 with n = 100 (approx using standard formula).

• A. (0.304, 0.496)
• B. (0.280, 0.520)
• C. (0.360, 0.440)
• D. (0.350, 0.450)

Answer: (A)

Estimating a proportion with a confidence interval using the normal approximation. For a proportion, the standard error is sqrt(p_hat(1-p_hat)/n). The 95% interval uses a z-score of 1.96. With p_hat = 0.40 and n = 100, the SE = sqrt(0.40×0.60/100) = sqrt(0.0024) ≈ 0.049. The margin of error is 1.96 × 0.049 ≈ 0.096. So the interval is 0.40 ± 0.096, i.e., (0.304, 0.496). This matches the shown result. Other options would require a larger or smaller margin of error than 0.096, or a different standard error, which is why they don’t fit.

Estimating a proportion with a confidence interval using the normal approximation. For a proportion, the standard error is sqrt(p_hat(1-p_hat)/n). The 95% interval uses a z-score of 1.96. With p_hat = 0.40 and n = 100, the SE = sqrt(0.40×0.60/100) = sqrt(0.0024) ≈ 0.049. The margin of error is 1.96 × 0.049 ≈ 0.096. So the interval is 0.40 ± 0.096, i.e., (0.304, 0.496). This matches the shown result. Other options would require a larger or smaller margin of error than 0.096, or a different standard error, which is why they don’t fit.