Why is the t-distribution used for a confidence interval for a mean when sigma is unknown?

Why is the t-distribution used for a confidence interval for a mean when sigma is unknown?

• A. Because it is symmetric
• B. Because it has lighter tails than the normal
• C. Because it accounts for the uncertainty in estimating sigma with s and depends on degrees of freedom.
• D. Because it does not depend on sample size

Answer: (C)

When sigma is unknown, we estimate variability from the data using s, the sample standard deviation. The statistic that standardizes the sample mean becomes (X̄ − μ) / (s/√n). Because s is itself a random quantity, this pivot does not follow the normal distribution exactly; instead it follows a t distribution with n−1 degrees of freedom. The t distribution has heavier tails than the normal, which reflects the extra uncertainty from estimating sigma with s. That is why the confidence interval for the mean uses critical values from the t distribution and depends on the degrees of freedom. As the sample size grows, s becomes a more precise estimate of sigma and the t distribution approaches the normal, so the interval looks more like the familiar z-interval. The key contrast is not about symmetry alone or about independence from sample size; the crucial point is the dependence on degrees of freedom due to estimating sigma with s, which is captured by the t distribution.

When sigma is unknown, we estimate variability from the data using s, the sample standard deviation. The statistic that standardizes the sample mean becomes (X̄ − μ) / (s/√n). Because s is itself a random quantity, this pivot does not follow the normal distribution exactly; instead it follows a t distribution with n−1 degrees of freedom. The t distribution has heavier tails than the normal, which reflects the extra uncertainty from estimating sigma with s. That is why the confidence interval for the mean uses critical values from the t distribution and depends on the degrees of freedom. As the sample size grows, s becomes a more precise estimate of sigma and the t distribution approaches the normal, so the interval looks more like the familiar z-interval. The key contrast is not about symmetry alone or about independence from sample size; the crucial point is the dependence on degrees of freedom due to estimating sigma with s, which is captured by the t distribution.