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

Definition of the sampling distribution of the $t$ statistic

Sampling distribution of $ t$:

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 drew many more samples. Specifically, suppose that we drew 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 would give different $ t$ values. The distribution of all these $ t$ values is the sampling distribution of $ t$. Note that this sampling distribution is purely hypothetical. We would never really draw an infinite number of samples, but hypothetically, we could.

Sampling distribution of $ t$ if H0 were true:

Suppose that the assumptions of the two sample $ t$ test hold, and that the null hypothesis that $\mu_1 = \mu_2$ is true. Then the sampling distribution of $ t$ is approximately the $ t$ distribution with $ k$ degrees of freedom (see the overview for possible values of $ k$). That is, most of the time we would find $ t$ values close to 0, and only sometimes we would find $ t$ values further away from 0. If we find a $ t$ value in our actual sample that is far away from 0, this is a rare event if the null hypothesis were true, and is therefore considered evidence against the null hypothesis ($ t$ value in rejection region, small $ p$ value).