Answer
a) Fail to reject the null hypothesis.
b)$\mu_1-\mu_2$ is between -2650.7 and 1557.3.
Work Step by Step
a) Null hypothesis:$\mu_1=\mu_2$, alternative hypothesis:$\mu_10.1. 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 0.1, hence we fail to reject the null hypothesis.
b) The corresponding critical value using the table: $t_{\alpha/2}=t_{0.005}=2.601.$ The margin of error: $E=t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=2.601\sqrt{\frac{8632.5^2}{186}+\frac{7301.2^2}{210}}=2104.2.$ Hence the confidence interval $\mu_1-\mu_2$ is between $\overline{x_1}-\overline{x_2}-E$=(15668.5-16215)-2104.2=-2650.7 and$\overline{x_1}-\overline{x_2}+E$=(15668.5-16215)+2104.2=1557.3.
