The time to deliver packaged containers by a logistics company is found from samples of size 4. The mean and standard deviation of delivery times is estimated to be 140 hours and 6 hours, respectively.
(a) Find the 2σ and 3σ control limits for the average delivery time.
(b) Explain a type I and type II error specifically in this context.
(c) Suppose that Rules 1 and 3 are used simultaneously to detect out-of-control conditions. Assuming independence of the rules, what is the overall probability of a type I error for 3σ control limits?
(d) If the mean delivery time shifts to 145 hours, using Rule 1, what is the probability of not detecting this by the second sample after the shift?
(e) What is the ARL using Rule 1? Explain.