Testing for Differences in Two Populations: Variances. After the firm in question (1) was inspected by the Dept. of Weights and Measures, a new random sample of forty observations on fill weights was taken the next day, resulting in the following summary statistics:

\(m^{\prime}{ }_{1}=16.02753\) and \(s^{2}=.9968 \times 10^{-2}\).

You may continue to assume normality for the population distribution.

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

(b) Repeat

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

(c) Define a GLR size .05 test of the \(H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}\) versus \(H_{a}\) : not \(H_{0}\). Either go to the computer and perform this test, or else define an approximation to this test based on "equal tails" and perform the test.

(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.