#### Answer

a) Fail to reject the null hypothesis.
b)$\mu_1-\mu_2$ is between -18.17 and 10.23.

#### Work Step by Step

a) Null hypothesis:$\mu_1=\mu_2$, alternative hypothesis:$\mu_1\ne\mu_2$. Hence the value of the test statistic: $t=\frac{(\overline{x_1}-\overline{x_2})-(\mu_1-\mu_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}=\frac{(70.29-74.26)-(0)}{\sqrt{22.09^2/30+18.15^2/32}}=-0.77.$ The degree of freedom: $min(n_1-1,n_2-1)=min(30-1,32-1)=29.$ The corresponding P-value by using the table: p is more than 0.2. If the P-value is less than $\alpha$, which is the significance level, then this means the rejection of the null hypothesis. Hence:P is more than $\alpha=0.01$, because it is more than 0.2, hence we fail to reject the null hypothesis.
b) The corresponding critical value using the table: $t_{\alpha/2}=t_{0.005}=2.756.$ The margin of error: $E=t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=2.756\sqrt{\frac{22.09^2}{30}+\frac{18.15^2}{32}}=14.2.$ Hence the confidence interval $\mu_1-\mu_2$ is between $\overline{x_1}-\overline{x_2}-E$=(70.29-74.26)-14.2=-18.17 and$\overline{x_1}-\overline{x_2}+E$=(70.29-74.26)+14.2=10.23.