Paired sample $t$ test - sampling distribution of the sample mean of the difference scores, and its standard error
Definition of the sampling distribution of the sample mean of the difference scores $\bar{y}$, and its standard error
Sampling distribution of the sample mean of the difference scores $ \bar{y}$:When we have a sample consisting of $ N$ difference scores (e.g., difference scores of pre- and post-measurements before and after an intervention), we can compute the sample mean of the difference scores: $ \bar{y}$. Now suppose that we repeated our study several times. Specifically, suppose that we repeated our study an infinite number of times, so we have an infinite number of samples of difference scores, each of size $ N$. In each sample, we could compute the sample mean of difference scores $ \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 repeat our study an infinite number of times, but hypothetically, we could. Standard error:Suppose that the assumptions of the paired 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 of the difference scores $\bar{y}$ is removed from $\mu_0$: the population mean of the difference scores according to the null hypothesis. |