##### $t$ and $z$ test for correlation: sampling distribution of $t$ and of $z$

Definition of the sampling distribution of the $t$ statistic and the $z$ statistic

##### Sampling distribution of $t$:

As you may know, when we test H0: $\rho = 0$, we compute the $t$ statistic $$t = \dfrac{r \times \sqrt{N - 2}}{\sqrt{1 - r^2}}$$ 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 $t = \frac{r \times \sqrt{N - 2}}{\sqrt{1 - r^2}}$. 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 $t$ test for the correlation hold, and that the null hypothesis that $\rho = 0$ is true. Then the sampling distribution of $t$ is the $t$ distribution with $N - 2$ 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 study 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).

##### Sampling distribution of $z$:

When we test H0 values other than $\rho = 0$, we compute the $z$ statistic $$z = \dfrac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\dfrac{1}{N - 3}}}$$ based on our sample data. Now suppose that we would draw many more samples. Specifically, suppose that we would draw an infinite number of samples, each of size $N$. In each sample, we could compute the $z$ statistic $z = \frac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\frac{1}{N - 3}}}$. Different samples would give different $z$ values. The distribution of all these $z$ values is the sampling distribution of $z$. 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 $z$ if H0 were true:

Suppose that the assumptions of the $z$ test for the correlation hold, and that the null hypothesis that $\rho = \rho_0$ is true. Then the sampling distribution of $z$ is approximately the normal distribution with mean 0 and standard deviation 1 (standard normal). That is, most of the time we would find $z$ values close to 0, and only sometimes we would find $z$ values further away from 0. If we find a $z$ value in our actual study 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 ($z$ value in rejection region, small $p$ value).