To increase business, the owner of a clothing store is running a promotion in which a customer's bill can be selected at random to receive a discount. When a customer's bill is printed, a program in the cash register determines randomly whether the customer will receive a discount on the bill. The program was written to generate a discount with a probability of 0.22; that is, 22% of the bills get a discount in the long run. However, the owner is concerned the program is incorrect and is not generating the intended long-run proportion of 0.22. The owner selects a random sample of bills and finds that only 17% of them received a discount. The confidence interval for p, the proportion of bills that will receive a discount in the long run, is 0.17 ± 0.05, and all conditions for inference are met.
Consider the confidence interval 0.17 ± 0.05.
Part A: Does the confidence interval provide convincing statistical evidence that the program is NOT working as intended? Justify your answer. (3 points)
Part B: Does the confidence interval provide convincing statistical evidence that the program generates the discount with a probability of 0.22? Justify your answer. (2 points)
Part C: A second random sample of bills is taken that is five times the size of the original sample. In the second sample, 17% of the bills received the discount. Determine the value of the margin of error based on the second sample of bills used to compute an interval for p with the same confidence level as that of the original interval. (2 points)
Part D: Based on the margin of error in part C obtained from the second sample, is the program working as intended? Justify your answer.