Test the claim below about the mean of the differences for a population of paired data at the level of significance $\alpha$. Assume the samples are random and dependent, and the populations are normally distributed. Claim: $\mu_{d}<0 ; \alpha=0.05$. Sample statistics: $\bar{d}=1.7, s_{d}=3.2, n=15$ Identify the null and alternative hypotheses. Choose the correct answer below. A. \[ \begin{array}{l} H_{0}: \mu_{d}>0 \\ H_{a}: \mu_{d} \leq 0 \end{array} \] C. \[ \begin{array}{l} H_{0}: \mu_{d}=0 \\ H_{a}: \mu_{d} \neq 0 \end{array} \] E. \[ \begin{array}{l} H_{0}: \mu_{d} \leq 0 \\ H_{a}: \mu_{d}>0 \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu_{d}<0 \\ H_{a}: \mu_{d} \geq 0 \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu_{d} \geq 0 \\ H_{a}: \mu_{d}<0 \end{array} \] F. \[ \begin{array}{l} H_{0}: \mu_{d} \neq 0 \\ H_{a}: \mu_{d}=0 \end{array} \]
Solution
Step 1 ：The null hypothesis (H0) is usually a statement of no effect or no difference. It is the hypothesis that the researcher is trying to disprove. The alternative hypothesis (Ha) is the hypothesis that the researcher wants to prove.
Step 2 ：In this case, the claim is that the mean of the differences for a population of paired data is less than 0. This claim is what we are trying to prove, so it should be the alternative hypothesis. The null hypothesis should be the opposite of the claim, that is, the mean of the differences for a population of paired data is greater than or equal to 0.
Step 3 ：Therefore, the null and alternative hypotheses should be: H0: μd ≥ 0, Ha: μd < 0
Step 4 ：So, the correct answer is D.
Step 5 ：Final Answer: \[\boxed{\begin{array}{l} H_{0}: \mu_{d} \geq 0 \\ H_{a}: \mu_{d}<0 \end{array}}\]
