We can compare the relationship of the t-distribution with n − 1 degrees of freedom with the normal distribution in several ways. As n gets large, the t-distribution converges to the standard normal. But what happens when n is “small,” and what do we mean by “large?”

A few comparative graphs can give us an idea. For n = 10 we can use boxplots of simulated data to examine the tails, or we can compare plots of theoretical quantiles or densities. These plots are created as follows:

Repeat the above for n = 3, 25, 50, and 100. What value of n seems “large” enough to say that the two distributions are essentially the same?

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