Testing for Differences in Two Populations: Means} (Equal Variances).
Referring to the data obtained in both preceding questions, consider testing whether the two populations means are the same. Assume that the population variances are identical in the two cases-did you find any evidence to contradict this in your answer to (2) above?

(a) Define a GLR size 05 test of the \(H_{0}: \mu_{1} \leq \mu_{2}\) versus \(H_{a}\) : not \(H_{0}\), where \(\mu_{1}\) and \(\mu_{2}\) refer to the means of the populations from which the first and second sample were taken. Test the hypothesis.

(b) Repeat

(a) for the hypothesis \(H_{0}: \mu_{1} \geq \mu_{2}\) versus \(H_{a}\) : not \(H_{0}\).

(c) Define a GLR size .05 test of the \(H_{0}: \mu_{1}=\mu_{2}\) versus \(H_{a}\) : not \(H_{0}\). Test the hypothesis at the 10 level of significance.

(d) Define the power function for each of the hypothesis testing procedures you defined above. With the aid of a computer, plot the power functions and interpret their meanings.