When considering the difference of two means, there are two common cases: the two samples are paired or they are independent. (There are instances where the data are neither paired nor independent.) The paired case was treated in Section 3.1, where the one-sample methods were applied to the differences from the paired observations. We examined the second and more complex scenario in this section.
When applying the normal model to the point estimate (corresponding to unpaired data), it is important to verify conditions before applying the inference framework using the normal model. First, each sample mean must meet the conditions for normality; these conditions are described in Chapter 2.6 on page 2.8.5. Secondly, the samples must be collected independently (e.g. not paired data). When these conditions are satisfied, the general inference tools of Chapter 2.6 may be applied.
For example, a confidence interval may take the following form:
When we compute the confidence interval for , the point estimate is the difference in sample means, the value corresponds to the confidence level, and the standard error is computed from Equation (3.1) on page 3.1. While the point estimate and standard error formulas change a little, the framework for a confidence interval stays the same. This is also true in hypothesis tests for differences of means.
In a hypothesis test, we apply the standard framework and use the specific formulas for the point estimate and standard error of a difference in two means. The test statistic represented by the Z score may be computed as
When assessing the difference in two means, the point estimate takes the form , and the standard error again takes the form of Equation (3.1) on page 3.1. Finally, the null value is the difference in sample means under the null hypothesis. Just as in Chapter 2.6, the test statistic is used to identify the p-value.