u/Anonymous_299912

Most people reason that the t-statistic should be used over the z-statistic, since the z-statistic requires the knowledge of the population's variance. I want to challenge this notion:

Let's call the arithmetic average of your random variable, X_bar. If you have determined your sample size to be small, then X_bar is not normally distributed. This is the Central Limit Theorem. If your random variable is not normally distributed, then you can't use the t-statistic.

It naturally follows that if you're assuming X_bar is normally distributed, then you are also assuming that your sample size is large. If your sample size is large, then the sample variance of your sample, with the correction, should reasonably equal the population variance.