Two-Population & Paired Analysis

Recap

• Goal: compare two populations to detect a real difference, not random variation.

Comparison of Two Population Means

• Independent samples from two distinct groups.

• rameter of interest: $\mu<em>1 - \mu</em>2$ (difference between population means).Pa

• Compute sample means $\bar{x}<em>1, \bar{x}</em>2$ and their difference.

• Build $95\%$ confidence interval (CI): $(\bar{x}<em>1-\bar{x}</em>2) \pm t^*\cdot SE$.

• Decision rule: if $0$ lies outside CI, conclude a significant difference.

Paired Differences

• Observations naturally paired (same subject measured twice or matched units).

• Create difference variable $d<em>i = x</em>i - y_i$.

• Reduces to single-sample problem on $\mu_d$ (mean of differences).

• CI: $\bar{d} \pm t^*\cdot SE_d$.

• Check if $0$ is in CI to infer significance.

Choosing the Method

• Two population means: independent groups, different subjects.

• Paired differences: same subjects, or matched pairs, measuring change.

Key Terms

• Difference between two means.

• Mean of the differences.

• Significance judged by whether $0$ belongs to the confidence interval.