Which of the following best describes the null hypothesis for an independent-samples t-test?

• A. The sample means are equal: ${ṅ}_1={ṅ}_2$
• B. The variances of the two populations are equal: ${\sigma }_1={\sigma }_2$
• C. The means of the two populations are not equal: ${\mu }_1\ne {\mu }_2$
• D. The means of the two populations are equal: ${\mu }_1={\mu }_2$

Answer: (D)

Verified step by step guidance

1. Understand that the null hypothesis (denoted as \(H_0\)) in an independent-samples t-test is a statement about the population parameters, not the sample statistics.
2. Recall that the independent-samples t-test is used to compare the means of two independent populations to see if there is evidence that they differ.
3. Identify that the null hypothesis typically states that there is no difference between the population means, which can be written as \(H_0: \mu_1 = \mu_2\).
4. Note that the alternative hypothesis (\(H_a\)) would state that the population means are not equal, i.e., \(H_a: \mu_1 \neq \mu_2\).
5. Recognize that statements about sample means (like \(\bar{x}_1 = \bar{x}_2\)) or population variances (like \(\sigma_1 = \sigma_2\)) are not the null hypothesis for the independent-samples t-test, although equality of variances is an assumption that may be tested separately.