Answer
a) Fail to reject the null hypothesis.
b)$\mu_1-\mu_2$ is between -0.19 and 0.61.
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: $t=\frac{(\overline{x_1}-\overline{x_2})-(\mu_1-\mu_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}=\frac{(98.38-98.17)-(0)}{\sqrt{0.45^2/15+0.65^2/91}}=1.559.$ The degree of freedom: $min(n_1-1,n_2-1)=min(15-1,91-1)=14.$ The corresponding P-value by using the table: p is between 0.05 and 0.1. 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 0.05, hence we fail to reject the null hypothesis.
b) The corresponding critical value using the table: $t_{\alpha/2}=t_{0.005}=2.977.$ The margin of error: $E=t_{\alpha/2}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=2.977\sqrt{\frac{0.45^2}{15}+\frac{0.65^2}{91}}=0.4.$ Hence the confidence interval $\mu_1-\mu_2$ is between $\overline{x_1}-\overline{x_2}-E$=(98.38-98.17)-0.4=-0.19 and$\overline{x_1}-\overline{x_2}+E$=(98.38-98.17)-0.4=0.61.
