Companies that sell food and other consumer products watch for problems. Occasional complaints from dissatisfied customers always occur, but an increase in the level of complaints may signal a serious problem that is better handled sooner than later. With food safety, that’s a particular concern. Getting out ahead of the bad news and actively working to fix a problem can save the reputation of a company.
The company in this exercise sells frozen, prepackaged dinners. It generally receives one or two calls to its problem center each month from consumers who say “Your food made me sick.” Usually, it’s not the food that is to blame, but some other cause like the flu. If the number of calls rises to a rate approaching six calls per month, however, then there’s a real problem. To be specific, let’s say that the normal rate of calls is 1.5 per month and that the rate of calls when there is a serious problem is 6 per month.
Motivation
(a) It is important for a company to react quickly when it recognizes that a product threatens public health. Why is it also important for a company not to overreact, acting as if there is a problem when in fact everything is operating normally?
(b) What are the trade-offs between reacting quickly to a real problem and overreacting when in fact there is no real problem?
Method
(c) What type of random variable seems suited to modeling the number of calls during a normal period? During a problem period?
(d) What assumptions are necessary for the use of the random variable chosen in part (c)? Do these seem reasonable in the context of this problem?
Mechanics
(e) During normal months, would it be surprising to receive more than 3 calls to the problem center?
(f) During problem months, would it be surprising to receive more than 3 calls to the problem center?
(g) The company seldom has problems, and management believes, without feedback from the problem center, that there is a 5% chance of a problem. If the company receives more than 3 calls to the problem center in a month, then what is the probability that there is in fact a problem?
Message
(h) More than 3 calls arrived at the problem center in the past month. Explain for management the chances that there is a problem.