Suppose two independent random samples are taken from two normal populations with unknown and unequal variances. Which statistical test is most appropriate for testing whether the population means are equal?

• A. Welch's $t$-test
• B. Pooled variance $t$-test
• C. Paired $t$-test
• D. $Z$-test for two means

Answer: (A)

Verified step by step guidance

1. Identify the key characteristics of the problem: two independent samples, normal populations, unknown and unequal variances.
2. Recall that when variances are unknown and assumed equal, the pooled variance t-test is appropriate; however, when variances are unequal, this assumption is violated.
3. Understand that the paired t-test is used for dependent or matched samples, which is not the case here since the samples are independent.
4. Recognize that the Z-test for two means requires known population variances or large sample sizes to approximate normality, which is not given here.
5. Conclude that Welch's t-test is designed specifically for comparing means from two independent samples when variances are unknown and unequal, making it the most appropriate test.