Confidence Intervals for Two Samples — Comparing Groups
Foundations of Statistics
Comparing Groups with Precision
Two-sample intervals estimate the difference between population parameters, providing the foundation for comparing treatments, groups, or conditions. They answer the practical question: how different are these groups really?
- A/B Testing — Quantifying the true difference in website conversion rates
- Clinical Trials — Estimating treatment versus control group differences
- Social Science — Measuring effect sizes in observational studies
The difference between groups is often more important than the groups themselves.
Core Concepts
Two-sample confidence intervals estimate the difference between two population parameters (means or proportions). They are the foundation for comparing groups.
Welch's t-Interval (Unequal Variance)
Derivation: The Two-Sample Pivot
Proof sketch: The numerator is normal with mean 0 and variance . Standardize: . By Cochran's theorem, , independent of and . Therefore where , giving by definition.
CI for Difference of Proportions
Worked Example: Clinical Trial
Two drugs are compared for blood pressure reduction. Drug A: , mmHg, . Drug B: , mmHg, . Construct a 95% CI for .
Step 1: Check if equal variance is reasonable: . The ratio is less than 2, so pooled t is reasonable.
Step 2: Compute pooled standard deviation:
Step 3: Compute standard error:
Step 4: Critical value: (interpolating from t-table).
Step 5: Construct CI: