Hypothesis Tests and Sampling Distribution for a Population Mean Difference

In short, if you want to test about a population mean difference, you just use a one-sample t-test for one mean. That's it!

There are still one-tailed tests and two-tailed tests.

The test statistic is still:

$$\text{Test statistic} = \frac{\text{Sample statistic - Null value}}{\text{Standard error}} = \frac{\bar{d} - 0}{\text{s.e.}(\bar{d})} = \frac{\bar{d}}{\frac{s_d}{\sqrt{n}}}$$

Where n is the number of pairs of observations. If \(H_0\) is true, this test has a \(t(n-1)\) distribution.

Required conditions:

• Random sample
• Normally distributed (the difference in means)

Ways to take a random sample:

• Taken separately from two populations
• One random sample is taken, and units are categorized as belonging to one population or another
• Participants are randomly assigned to one of two treatment conditions

If two populations are normally distributed, then \(\bar{x_1} - \bar{x_2}\) is approximately:

$$N(\mu_1 - \mu_2, \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$$