# $z$ test for a single proportion: sampling distribution of the $z$ statistic

Definition of the sampling distribution of the $z$ statistic

As you may know, when we perform a $ z$ test for a single proportion, we compute the $ z$ statistic
$$
z = \dfrac{p - \pi_0}{\sqrt{\dfrac{\pi_0(1 - \pi_0)}{N}}}
$$
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{p - \pi_0}{\sqrt{\frac{\pi_0(1 - \pi_0)}{N}}}$. Different samples will give different $ z$ values. The distribution of all these $ z$ values is the |