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: ${\mu }_d\ge 0;\alpha =0.10$. Sample statistics: $\overline{d}=-2.1,s_d=1.3,n=20$

Identify the null hypothesis by writing its complement.
A.
$$\begin{array}{l}{H}_0:{\mu }_d\ne 0\\ H_a:{\mu }_d=0\end{array}$$
C.
$$\begin{array}{l}{H}_0:{\mu }_d\ge 0\\ H_a:{\mu }_d<0\end{array}$$
E.
$$\begin{array}{l}{H}_0:{\mu }_d\le 0\\ H_a:{\mu }_d>0\end{array}$$
B.
$$\begin{array}{l}{H}_0:{\mu }_d>0\\ H_a:{\mu }_d\le 0\end{array}$$
D.
$$\begin{array}{l}{H}_0:{\mu }_d=0\\ H_a:{\mu }_d\ne 0\end{array}$$
F.
$$\begin{array}{l}{H}_0:{\mu }_d<0\\ H_a:{\mu }_d\ge 0\end{array}$$

The test statistic is $\mathrm{t}=-7.22$.
(Round to two decimal places as needed.)
The critical value(s) is(are) $t_0=◻$.
(Round to two decimal places as needed. Use a comma to separate answers as needed.)

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., ${\mu }_d\ge 0$. The level of significance, $\alpha$, is 0.10. The sample statistics are: $\overline{d}=-2.1$, $s_d=1.3$, and $n=20$.

Step 2 ：The null hypothesis, $H_0$, and its complement, $H_a$, are identified as follows: $H_0:{\mu }_d\ge 0$ 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 $H_0:{\mu }_d\ge 0$ and $H_a:{\mu }_d<0$. The critical value is $\boxed{{\displaystyle 1.33}}$.