Discussion
Suppose \(t\) test is applied to test the equality of means of two random variables X and Y, where X and Y are independent. Then :
Answer: A
For the t-test, the distributions of both X and Y must be normal with equal variances.
The two-sample t statistic \((\bar{X} - \bar{Y}) / (s_{p}\sqrt{1/n_{1} + 1/n_{2}})\) has an exact t distribution with \(n_{1} + n_{2} - 2\) degrees of freedom only when both populations are normal and share a common variance, so that the pooled estimate \(s_{p}^{2}\) is valid.
Normality alone (B) is not enough, because with unequal variances the statistic is not exactly t (the Behrens-Fisher problem). Equal variances alone (D) is not enough without normality, and C drops the normality assumption, which the exact test needs.
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