Regarding the assumptions and properties of the independent samples t-test, all of the following are correct EXCEPT:

Explanation

## Correct Answer Analysis **High-Yield:** The independent samples t-test has specific parametric assumptions that must be met or reasonably satisfied for valid inference. ### Key Assumptions of the t-test | Assumption | Requirement | Consequence if Violated | |---|---|---| | **Normality** | Data should be approximately normally distributed | Robust with large samples (CLT); use non-parametric test (Mann-Whitney U) if n < 30 and severely non-normal | | **Homogeneity of variance** | Variances in both groups should be equal | Use Welch's t-test (does not assume equal variances) | | **Independence** | Observations must be independent | Invalidates the test; use paired t-test if data are paired | | **Continuous data** | Data should be continuous or at least interval/ratio scale | Categorical data require chi-square or Fisher's exact test | ### Why Option 2 (Correct Answer) is Wrong **Key Point:** Option 2 states that homogeneity of variance is **mandatory**. This is **incorrect**. While equal variances is an *ideal* assumption, it is **not mandatory**: - If Levene's test shows unequal variances, **Welch's t-test** (which does not assume equal variances) is the appropriate alternative. - Welch's t-test is more conservative and uses adjusted degrees of freedom. - Modern statistical practice recognizes that the t-test is reasonably robust to moderate violations of the equal variance assumption, especially with equal or near-equal sample sizes. **Warning:** Saying an assumption is "mandatory" is a common trap in exam questions. The t-test is *robust* to many violations; it does not require *perfect* adherence to all assumptions. ### Why the Other Options Are Correct 1. **Option 0 (Normality & CLT):** Correct. With large samples (n > 30), the sampling distribution of the mean becomes approximately normal even if the underlying data are not, due to the Central Limit Theorem. 2. **Option 1 (Welch's t-test):** Correct. When variances are unequal (confirmed by Levene's test), Welch's t-test is the appropriate choice. 3. **Option 3 (Degrees of freedom):** Correct. The t-statistic under the null hypothesis follows a t-distribution with df = n₁ + n₂ − 2 for the standard t-test (or adjusted df for Welch's t-test). [cite:Park 26e Ch 12]