When estimating population standard deviation using "s," how is the margin of error computed?

The margin of error in estimating the population standard deviation using sample standard deviation "s" is computed using the t distribution primarily when the sample size is small (typically n < 30) and when the population standard deviation is unknown. This is because the t distribution accounts for the increased variability that might arise from estimating the population parameters from a smaller sample size.

As the sample size increases, the t distribution approaches the normal distribution, but for smaller samples, using the t distribution provides a more accurate estimate of the margin of error, reflecting the uncertainty involved in the estimation process. It incorporates the degrees of freedom from the sample size, which adjusts the confidence interval wider or narrower depending on the size, thereby giving a more reliable estimate when the sample size is small.

In contrast, while the normal distribution and Z distribution might be used for larger sample sizes or when the population standard deviation is known, they do not account for the added variability present in smaller samples. The binomial distribution is not applicable in this context, as it is used for discrete outcomes rather than for estimating population parameters such as standard deviation.