Answer
a) Reject the null hypothesis.
b)$\mu_1-\mu_2$ is between -25.54 and -3.1.
Work Step by Step
a) Null hypothesis:$\mu_1=\mu_2$, alternative hypothesis:$\mu_1$ is less than $\mu_2$. Hence the value of the test statistic: $t=\frac{(\overline{x_1}-\overline{x_2})-(\mu_1-\mu_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}=\frac{(74.3-88.62)-(0)}{\sqrt{12.87^2/30+22.09^2/32}}=-3.142.$ 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 less than 0.005. 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 less than $\alpha=0.01$, because it is less than 0.005, hence we reject the null hypothesis.
b) The corresponding critical value using the table: $t_{\alpha/2}=t_{0.01}=2.462.$ The margin of error: $E=t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=2.462\sqrt{\frac{12.87^2}{30}+\frac{22.09^2}{32}}=11.22.$ Hence the confidence interval $\mu_1-\mu_2$ is between $\overline{x_1}-\overline{x_2}-E$=(74.3-88.62)-11.22=-25.54 and$\overline{x_1}-\overline{x_2}+E$=(74.3-88.62)+11.22=-3.1.
