# Two sample $z$ test: sampling distribution of the $z$ statistic

Definition of the sampling distribution of the $z$ statistic

As you may know, when we perform a two sample $ z$ test, we compute the $ z$ statistic
$$
z = \dfrac{\bar{y}_1 - \bar{y}_2}{\sqrt{\dfrac{\sigma^2_1}{n_1} + \dfrac{\sigma^2_2}{n_2}}}
$$
based on our group 1 and group 2 samples. Now suppose that we would draw many more samples. Specifically, suppose that we would draw 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 $ z$ statistic $ z = \frac{\bar{y}_1 - \bar{y}_2}{\sqrt{\frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}}}$. Different samples will give different $ z$ values. The distribution of all these $ z$ values is the |