Which of the following is a required assumption for using the two-sample test of means?

• A. The samples must be from the same $\mathrm{population}$ and the $\mathrm{variances}$ must be unknown.
• B. The $\mathrm{populations}$ must be non-normal and the $\mathrm{sample}$ sizes must be less than $10$.
• C. The two $\mathrm{samples}$ must be dependent and drawn from $\mathrm{populations}$ with different $\mathrm{variances}$.
• D. The two $\mathrm{samples}$ are independent and both $\mathrm{populations}$ are normally distributed with equal $\mathrm{variances}$.

Answer: (D)

Verified step by step guidance

1. Understand that the two-sample test of means is used to compare the means of two independent groups to see if there is a statistically significant difference between them.
2. Recognize that one key assumption is that the two samples must be independent, meaning the selection or measurement of one sample does not affect the other.
3. Know that both populations from which the samples are drawn should be approximately normally distributed, especially important when sample sizes are small, to justify the use of the test.
4. Be aware that the test assumes equal variances (homogeneity of variance) in the two populations, which means the spread or variability in both groups should be similar.
5. Summarize that these assumptions ensure the validity of the test results: independence of samples, normality of populations, and equal population variances.