A complaint has been lodged against a major domestic manufacturer of potato chips stating that their \(16 \mathrm{oz}\) bags of chips are being underfilled. The manufacturer claims that their filling process produces fill weights that are normally distributed with a mean of \(16.1 \mathrm{oz}\) and a standard deviation of \(\leq .05\) so that over 97 percent of their product has a weight of \(\geq 16 \mathrm{oz}\). They suggest that their product be randomly sampled and their claims be tested for accuracy. Two independent random samples of observations on fill weights, each of size 250, resulted in the following summary statistics \(\bar{X}_{1}=16.05, \bar{x}_{2}=16.11\), \(s_{1}^{2}=.0016\), and \(s_{2}^{2}=.0036\).

(a) Define a UMPU level 05 test of \(H_{0}: \mu=16.1\) versus \(H_{a}: \mu eq 16.1\) based on a random sample of size 250 . Test the hypothesis using the statistics associated with the first random sample outcome. Plot and interpret the power function of this test.

(b) Define a UMPU level .05 test of \(H_{0}: \sigma \leq .05\) versus \(H_{a}: \sigma>.05\) based on a random sample of size 250 . Test the hypothesis using the statistics associated with the second random sample outcome. Plot and interpret the power function of this test.

(c) Calculate and interpret the \(p\)-values of the tests in

(a) and (b). (Hint: It might be useful to consider Bonferroni's inequality for placing an upper bound on the probability of Type I Error.)

(d) Treating the hypotheses in

(a) and

(b) as a joint hypothesis on the parameter vector of the normal population distribution, what is the probability of Type I Error for the joint hypothesis \(H_{0}: \mu=16.1\) and \(\sigma \leq .05\) when using the outcome of the two test statistics above to determine acceptance or rejection of the joint null hypothesis? Does the complaint against the company appear to be valid?

(e) Repeat (a-c) using a pooled sample of 500 observations.