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: $\mu_{d} \geq 0 ; \alpha=0.10$. Sample statistics: $\bar{d}=-2.1, s_{d}=1.3, n=20$ Identify the null hypothesis by writing its complement. A. \[ \begin{array}{l} H_{0}: \mu_{d} \neq 0 \\ H_{a}: \mu_{d}=0 \end{array} \] C. \[ \begin{array}{l} H_{0}: \mu_{d} \geq 0 \\ H_{a}: \mu_{d}<0 \end{array} \] E. \[ \begin{array}{l} H_{0}: \mu_{d} \leq 0 \\ H_{a}: \mu_{d}>0 \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu_{d}>0 \\ H_{a}: \mu_{d} \leq 0 \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu_{d}=0 \\ H_{a}: \mu_{d} \neq 0 \end{array} \] F. \[ \begin{array}{l} H_{0}: \mu_{d}<0 \\ H_{a}: \mu_{d} \geq 0 \end{array} \] The test statistic is $\mathrm{t}=-7.22$. (Round to two decimal places as needed.) The critical value(s) is(are) $t_{0}=\square$. (Round to two decimal places as needed. Use a comma to separate answers as needed.)
Solution
Step 1 ：The claim is that the mean of the differences for a population of paired data is greater than or equal to 0, i.e., \( \mu_{d} \geq 0 \). The level of significance, \( \alpha \), is 0.10. The sample statistics are: \( \bar{d} = -2.1 \), \( s_{d} = 1.3 \), and \( n = 20 \).
Step 2 ：The null hypothesis, \( H_{0} \), and its complement, \( H_{a} \), are identified as follows: \( H_{0}: \mu_{d} \geq 0 \) and \( H_{a}: \mu_{d} < 0 \).
Step 3 ：The test statistic is \( t = -7.22 \).
Step 4 ：The critical value, \( t_{0} \), is calculated using the given level of significance, \( \alpha = 0.10 \), and the degrees of freedom, \( df = n - 1 = 20 - 1 = 19 \).
Step 5 ：The critical value, \( t_{0} \), is approximately 1.33.
Step 6 ：Final Answer: The null hypothesis and its complement are \( H_{0}: \mu_{d} \geq 0 \) and \( H_{a}: \mu_{d} < 0 \). The critical value is \( \boxed{1.33} \).
