How do you construct a 95% confidence interval for a population mean when σ is known?

How do you construct a 95% confidence interval for a population mean when σ is known?

• A. x_bar ± t_{n-1} (s/√n)
• B. x_bar ± z0.975 * sqrt(p_hat*(1-p_hat)/n)
• C. x_bar ± z0.95 * (sigma/√n)
• D. x_bar ± z0.975 * (sigma/√n)

Answer: (D)

When sigma is known, the sample mean is normally distributed around the true mean with standard error sigma divided by the square root of the sample size. To form a 95% interval, you use the standard normal critical value that leaves 2.5% in each tail, which is z_{0.975} ≈ 1.96. So the interval is x̄ ± 1.96*(sigma/√n). This uses the known sigma to determine the margin of error directly from the normal distribution. The t-based form with s is used when sigma is unknown. The option with p̂ applies to proportions, not means. Using z with 0.95 would correspond to a 90% interval, not 95%, so that one isn’t appropriate for a 95% CI.

When sigma is known, the sample mean is normally distributed around the true mean with standard error sigma divided by the square root of the sample size. To form a 95% interval, you use the standard normal critical value that leaves 2.5% in each tail, which is z_{0.975} ≈ 1.96. So the interval is x̄ ± 1.96*(sigma/√n). This uses the known sigma to determine the margin of error directly from the normal distribution.

The t-based form with s is used when sigma is unknown. The option with p̂ applies to proportions, not means. Using z with 0.95 would correspond to a 90% interval, not 95%, so that one isn’t appropriate for a 95% CI.