When the population standard deviation is unknown, which formula is used to form the confidence interval for the mean?

When the population standard deviation is unknown, which formula is used to form the confidence interval for the mean?

• A. xbar ± t_{n-1,0.975} * s / sqrt(n)
• B. xbar ± z_{0.975} * s / sqrt(n)
• C. xbar ± t_{n,0.975} * s / sqrt(n)
• D. xbar ± t_{n-2,0.975} * s / sqrt(n)

Answer: (A)

When the population standard deviation is unknown, we form the confidence interval for the mean using the t distribution with degrees of freedom equal to n − 1, and we estimate the standard error with the sample standard deviation. The interval is centered at the sample mean and uses s/√n as the standard error, with the appropriate t critical value for a two-sided 95% interval. This gives x-bar ± t_{n−1,0.975} · (s/√n). The reason this is correct is that the t distribution accounts for extra uncertainty from estimating sigma with s, and the degrees of freedom reflect the n observations minus one estimate used. Using a z value would assume sigma is known, and using the wrong degrees of freedom would misstate the spread.

When the population standard deviation is unknown, we form the confidence interval for the mean using the t distribution with degrees of freedom equal to n − 1, and we estimate the standard error with the sample standard deviation. The interval is centered at the sample mean and uses s/√n as the standard error, with the appropriate t critical value for a two-sided 95% interval. This gives x-bar ± t_{n−1,0.975} · (s/√n). The reason this is correct is that the t distribution accounts for extra uncertainty from estimating sigma with s, and the degrees of freedom reflect the n observations minus one estimate used. Using a z value would assume sigma is known, and using the wrong degrees of freedom would misstate the spread.