##### One sample $t$ test - sampling distribution of the sample mean and its standard error

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

##### 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 error:

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

• The variable $y$ is normally distributed in the population, with mean $\mu$ and standard deviation $\sigma$
• 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 $\sigma / \sqrt{N}$. Since the one sample $t$ test does not make the assumption that the value of $\sigma$ is known (like the $z$ test does), we need to:
• estimate $\sigma$ with $s$: the standard deviation of $y$ in the sample
• estimate $\sigma / \sqrt{N}$ with $s / \sqrt{N}$
That is, we estimate the standard deviation of the sampling distribution of $\bar{y}$, $\sigma / \sqrt{N}$, with $s / \sqrt{N}$. We call this estimated standard deviation of the sampling distribution of $\bar{y}$ the standard error of $\bar{y}$.

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