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 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 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:
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. |