Chapter 10: Comparing Two Populations or Treatments

The fundamental distinction between independent samples, where separate individuals comprise each group, and paired data, where observations are matched or repeated within subjects, determines which inferential method applies. When comparing two population proportions from independent samples, students learn to construct confidence intervals and conduct two-sample z-tests for proportions, calculating standard errors that account for variability in both groups and verifying conditions such as adequate sample sizes and random selection. The chapter then addresses comparing two population means, introducing the two-sample t-test for independent samples when population standard deviations are unknown, emphasizing how the test statistic follows a t-distribution and requires checking normality and equality of variance assumptions. For paired data structures—including before-and-after designs, matched pairs studies, and repeated measures on the same subjects—the paired t-test provides the appropriate alternative by reducing the analysis to a single sample of differences. Throughout, the chapter stresses the critical importance of checking conditions for inference: randomness in data collection, approximate normality of sampling distributions, and independence of observations both within and between samples. Interpretation requires connecting statistical significance determined through p-values and confidence intervals to the practical context and real-world meaning of observed differences. The chapter explicitly addresses common mistakes, particularly mistakenly applying two-sample methods to paired data or ignoring the distinction between statistical significance and practical importance. By completing this chapter, students develop competence in selecting the correct inferential procedure, verifying assumptions, calculating and interpreting confidence intervals and test statistics using technology, and drawing conclusions that appropriately reflect both the mathematical evidence and the actual context of the problem.