One sample $z$ test - sampling distribution of the sample mean and its standard deviation

Definition of the sampling distribution of the sample mean $\bar{y}$ and its standard deviation

Sampling distribution of the sample mean $\bar{y}$:

When we draw a sample of size $N$ from the population, we can compute the sample mean of a variable $y$: $\bar{y}$. 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 sample mean $\bar{y}$. Different samples would give different sample means. The distribution of all these sample means is the sampling distribution of $\bar{y}$. Note that this sampling distribution is purely hypothetical. We would never really draw an infinite number of samples, but hypothetically, we could.

Standard deviation:

Suppose that the assumptions of the one sample $z$ test hold:

• The variable $y$ is normally distributed in the population, with mean $\mu$ and standard deviation $\sigma$
• The population standard deviation $\sigma$ is known
• The sample is a simple random sample from the population. That is, observations are independent of one another
Then the sampling distribution of $\bar{y}$ is normal with mean $\mu$ and standard deviation $\sigma / \sqrt{N}$.

Note that the $z$ statistic $z = \frac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$ thus indicates how many standard deviations $\sigma / \sqrt{N}$ the observed sample mean $\bar{y}$ is removed from $\mu_0$: the population mean according to the null hypothesis.