A clinical trial compares the mean reduction in systolic blood pressure (mmHg) between two antihypertensive drugs in 35 hypertensive patients randomized to Drug X and 38 patients randomized to Drug Y. The data are normally distributed with unequal variances. Which test best discriminates whether the mean blood pressure reduction differs significantly between the two drugs?

## Selecting the Correct t-Test Variant ### Data Characteristics **Key Point:** - Outcome: continuous (systolic BP reduction in mmHg) - Groups: two independent samples (Drug X vs. Drug Y) - Distribution: normal (parametric test appropriate) - Variances: **unequal** (critical discriminator) ### Why Welch's t-Test? **High-Yield:** Welch's t-test (also called unequal variance t-test or Welch–Aspin test) is the appropriate choice when: 1. Comparing means of two independent groups 2. Data are approximately normally distributed 3. Variances are unequal (heteroscedasticity) 4. Sample sizes may be unequal (35 vs. 38 here) ### Comparison of t-Test Variants | Feature | Student's t-test | Welch's t-test | |---------|------------------|----------------| | Variances | Equal | Unequal | | Assumption | Homogeneity of variance | No homogeneity required | | Robustness | Sensitive to variance inequality | Robust to variance inequality | | Degrees of freedom | n₁ + n₂ − 2 | Adjusted (Welch–Satterthwaite) | | When to use | Variances similar | Variances differ | **Clinical Pearl:** Levene's test is used to assess equality of variances. If p < 0.05, variances are unequal → use Welch's t-test. ### Welch's t-Test Formula $$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$ where $s_1^2$ and $s_2^2$ are unequal sample variances. **Mnemonic:** **UNEQUAL VARIANCES → WELCH'S t** — When variances differ, Welch corrects the degrees of freedom and standard error.