A certain business uses national telephone solicitation to sell its product. Its sales staff have individual weekly sales quotas of 10 sales that they must meet or else their job performance is considered to be unsatisfactory and they receive only base pay and no sales commission. In hiring sales staff, the company has claimed that the proportion of customers solicited that will ultimately buy the company's product is .05 , so that on average, 200 phone calls per week should produce the required 10 sales. The company requires that a salesperson keep a record of how many phone solicitations were made, and when the 10th sale is made, the salesperson must indicate the number of phone calls that were made to obtain the 10 sales. The data on the last 200 weekly quotas that were met by various salespersons indicated that 289 phone calls were needed, on average, to meet the quota. A disgruntled employee claims that the company has overstated the market for the product, and wants the quota lowered.

(a) Define a UMP level \(\alpha\) test of the hypothesis \(H_{0}\) : \(p=.05\) versus \(H_{a}: p<.05\). You may use an asymptotic normal distribution for the test statistic, if it has one.

(b) Test the hypothesis with a UMP size .10 test. Does the disgruntled employee have a legitimate concern?

(c) Examine the asymptotic power function of the test (i.e., construct a power function based on the asymptotic normal distribution of the test statistic). Interpret the implications of the power function for the test you performed, both from the perspective of the company and from the perspective of the employee. If you were primarily interested in worker's rights, might you design the test differently and/or would you consider testing a different null hypothesis? Explain.

(d) Suppose there was a substantial difference in the abilities of salespersons to persuade consumers to purchase the company's product. Would this have an impact on your statistical analysis above? Explain.