Why is a t distribution preferred when calculating confidence intervals for a sample where σ is unknown?

The t distribution is preferred for calculating confidence intervals when the population standard deviation (σ) is unknown primarily because it accounts for additional uncertainty that arises from estimating the standard deviation from the sample itself. When σ is unknown, using the sample standard deviation instead introduces variability, especially in small samples. The t distribution is more spread out and has heavier tails compared to the standard normal distribution (z distribution), which means it provides a more conservative estimate of the confidence interval. This is particularly beneficial when dealing with small sample sizes, where the sample mean might not be a precise estimator of the population mean. As sample sizes increase, the t distribution converges to the normal distribution, but for smaller samples, using the t distribution helps to ensure that we maintain the appropriate level of confidence in our interval estimates.