# Two sample $t$ test: sampling distribution of the $t$ statistic

Definition of the sampling distribution of the $t$ statistic

As you may know, when we perform a two sample $ t$ test (not assuming equal population variances), we compute the $ t$ statistic
$$
t = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}}}
$$
based on our group 1 and group 2 samples. Now suppose that we would draw many more samples. Specifically, suppose that we would draw an infinite number of group 1 and group 2 samples, each time of size $ n_1$ and $ n_2$. Each time we have a group 1 and group 2 sample, we could compute the $ t$ statistic $ t = \frac{\bar{y}_1 - \bar{y}_2}{\sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}}$. Different samples will give different $ t$ values. The distribution of all these $ t$ values is the |