One 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 one sample $ t$ test, we compute the $ t$ statistic
$$
t = \dfrac{\bar{y} - \mu_0}{s / \sqrt{N}}
$$
based on our sample data. Now suppose that we drew many more samples. Specifically, suppose that we drew an infinite number of samples, each of size $ N$. In each sample, we could compute the $ t$ statistic $ t = \frac{\bar{y} - \mu_0}{s / \sqrt{N}}$. 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 one sample $ t$ test hold, and that the null hypothesis that $\mu = \mu_0$ is true. Then the sampling distribution of $ t$ is the $ t$ distribution with $ N - 1$ degrees of freedom. 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).