We can investigate how robust the T-statistic is to changes in the underlying parent population from normality. In particular, we can verify that if the parent population is not too skewed or is symmetric without too heavy a tail then the T-statistic will still have the t-distribution for its sampling distribution. (This is for small samples where n is not so large that the central limit theorem is applicable.)
A simulation of the T-statistic when i are Normal(0, 1) may be done as follows:
The quantile-quantile plot compares the distribution of the sample with a sample from the t-distribution. If the above is executed, it should produce a graphic with points that are close to linear, as the sampling distribution is the t-distribution.
To test different parent populations you can change the line x