For a 95% confidence interval for the population mean, if the population standard deviation is known, the standard normal distribution should be used.

For a 95% confidence interval for the population mean, if the population standard deviation is known, the standard normal distribution should be used.

• A. True
• B. False
• C. Both
• D. Not sure

Answer: (A)

When the population standard deviation is known, you use the standard normal distribution to form a confidence interval for the population mean because the sampling distribution of the sample mean is Normal with standard error σ/√n. This allows you to use the z critical value (for 95% confidence, about 1.96) in the interval x̄ ± z* (σ/√n). In other words, the 95% CI is x̄ ± 1.96 × σ/√n. If σ were unknown, you’d replace σ with the sample standard deviation and use the t distribution with df = n − 1, yielding x̄ ± t_{n−1,0.025} × s/√n. The emphasis here is that known σ → normal (z) methods; unknown σ → t methods.

When the population standard deviation is known, you use the standard normal distribution to form a confidence interval for the population mean because the sampling distribution of the sample mean is Normal with standard error σ/√n. This allows you to use the z critical value (for 95% confidence, about 1.96) in the interval x̄ ± z* (σ/√n). In other words, the 95% CI is x̄ ± 1.96 × σ/√n.

If σ were unknown, you’d replace σ with the sample standard deviation and use the t distribution with df = n − 1, yielding x̄ ± t_{n−1,0.025} × s/√n. The emphasis here is that known σ → normal (z) methods; unknown σ → t methods.