Let X be the number of defective items found by a quality inspector in a random batch of 15 items from a particular manufacturing process. The probability distribution of X is provided in the file S04_18.xlsx. This firm earns $500 profit from the sale of each acceptable item in a given batch. In the event that an item is found to be defective, it must be reworked at a cost of $100 before it can be sold, thus reducing its per-unit profit to $400.
a. Find the mean and standard deviation of the profit earned from the sale of all items in a given batch.
b. What is the probability that the profit earned from the sale of all items in a given batch is within two standard deviations of the mean profit level? Is this result consistent with the empirical rules from Chapter 2? Explain.