a. Recalling that the fill weight and R charts are based on subgroups of size 5, and assuming that the fill weights are approximately normally distributed, calculate the natural tolerance limits for the process.
b. Suppose that management wishes to reduce the mean fill weight in order to save money by "giving away" less product. However, since customers expect each bag to contain at least 50 pounds of product, management wishes to leave some process leeway. Therefore, after the mean fill weight is reduced, the lower natural tolerance limit is to be no less than 50.5 lb. Based on the natural tolerance limits, how much can the mean fill weight be reduced? II the product costs $2 per pound, and if 1 million bags are sold per year, what is the yearly cost reduction achieved by lowering the mean till weight?