##### $z$ and $t$ 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 would draw many more samples. Specifically, suppose that we would repeat our study an infinite number of times. In each of the studies, we could compute the $t$ statistic $t = \frac{r \times \sqrt{N - 2}}{\sqrt{1 - r^2}}$ based on the sampled data. Different studies would be based on different samples, resulting in 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 will never really repeat our study an infite number of times, 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 repeat our study an infinite number of times. In each of the studies, we could compute the $z$ statistic $z = \frac{r_{Fisher} - \rho_{0_{Fisher}}}{\sqrt{\frac{1}{N - 3}}}$ based on the sampled data. Different studies would be based on different samples, resulting in 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 will never really repeat our study an infite number of times, 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).