Answer
a) Reject the null hypothesis.
b)$\mu_1-\mu_2$ is between -8.13 and -1.47.
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: $t=\frac{(\overline{x_1}-\overline{x_2})-(\mu_1-\mu_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}=\frac{(131.37-136.17)-(0)}{\sqrt{5.13^2/30+5.35^2/30}}=-3.547.$ The degree of freedom: $min(n_1-1,n_2-1)=min(30-1,30-1)=29.$ 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: $E=t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=2.462\sqrt{\frac{5.13^2}{30}+\frac{5.35^2}{30}}=3.33.$ Hence the confidence interval $\mu_1-\mu_2$ is between $\overline{x_1}-\overline{x_2}-E$=(131.37-136.17)-3.33=-8.13 and$\overline{x_1}-\overline{x_2}+E$=(131.37-136.17)+3.33=-1.47.
