$t$ 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 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 $ 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).

t distribution

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).

Standard normal distribution