Control Charting: A large mail-order house has initiated a quality control program. They randomly sample 100 of each day's orders and monitor whether or not the mail order-taking process is "under control" in the sense that errors in order-taking are at minimum levels. The number of orders per day is sufficiently large that one can assume that the sampling is done with replacement. The daily error proportion prior to the initiation of this program has been 3.2 percent.

(a) Define a UMPU level \(\alpha\) test of the hypothesis \(H_{0}\) : \(p=.032\) versus \(H_{a}: p eq .032\). You may use the asymptotic normal distribution of the test statistic in defining the critical region.

(b) Conduct a size . 05 UMPU test of the null hypothesis on a day where \(\bar{x}=3.4\). Is the order process under control?

(c) After quality control training and closer monitoring of clerical workers, a daily random sample resulted in \(\bar{X}=2.6\). Is there evidence that quality has increased over what it has been in the past? Why or why not?

(The mail order company of Alden's Inc. was one of the earliest companies to use control charting techniques for monitoring clerical work. See Neter, John, (1952), "Some Applications of Statistics for Auditing", Journal of the American Statistical Association, March, pp. 6-24.