STAT 301 - Business Statistics
Class Activity – Normal Distribution
(counts as Lecture Review #8)

This activity asks you to use basic quantitative techniques to help you make a business decision. The situation is fictional but realistic. Work together with your group members to answer these questions. Your group should submit a single write-up with your analyses, at the end of the period. If you are not able to complete the activity during class time, then each group member may submit an individual write-up at the beginning of next class.

"Chocolate Frosted Sugar Bombs" is the name of a popular breakfast cereal manufactured by the General Junkfoods Corporation. The cereal is produced and packaged at the company’s factory in the town of Bean Blossom, Indiana. Automated machinery is used to fill individual boxes with cereal. No machine is perfect, of course, and so the amount of cereal actually in a box will vary slightly from box to box. In fact, the amount of cereal in the box (unsurprisingly) tends to follow a normal distribution.

General Junkfoods fell afoul of the Federal Trade Commission (FTC) several years ago for false advertising. (Their "20 ounce" boxes of cereal actually contained considerably less than that amount.) As part of the settlement reached with the FTC, General Junkfoods has agreed to undergo an inspection plan conducted by an independent product testing organization. Under the terms of the agreement, the testing organization will randomly select 1000 boxes of cereal annually from the company’s production. (The organization is given complete freedom of the production floor, in order to obtain this random sample apart from any company meddling.) The contents of each of these boxes will be weighed at the testing firm’s laboratories under carefully controlled conditions. For every box in the sample that weighs less than the advertised 20 ounces, General Junkfoods will incur a $250,000 fine. (Thus, if the lab finds 10 of the 1000 boxes in violation, this means $2,500,000 in fines for General Junkfoods.)

"Chocolate Frosted Sugar Bombs" is a very popular breakfast cereal, and General Junkfoods finds itself in the enviable position of being able to sell every box that it can produce, at an average profit of $0.75 per box. That’s the good news. The bad news is that the manufacturing plant is badly out of date and is already operating at full capacity, which is 42,000 tons of cereal per year. (That’s a lot of "20 ounce" boxes, however.) Moreover, General Junkfoods does not anticipate being able to afford to expand capacity in the near future.

The automated machinery used to fill the boxes can be adjusted somewhat, to produce a specified mean amount of cereal per box. At present, the machinery is set to fill boxes to an average of 20.5 ounces. (Yes, this is more than the advertised "20 ounces." But this is standard in the industry. You must meet the advertised capacity claims — not necessarily every time, but at least with reliable regularity. Besides, General Junkfoods cannot afford too many $250,000 fines for underfilled boxes.) The standard deviation of the machinery is NOT adjustable. It is a function of the underlying design and engineering of the equipment, and has been stable at 0.2 ounces for the past several years.

QUESTION 1:

Currently, General Junkfoods puts an average of 20.5 ounces of cereal in each box of Chocolate Frosted Sugar Bombs. Use the information in the problem description to answer the following questions.

a) How many boxes of cereal is the company able to produce annually, if each contains 20.5 ounces of cereal on average?

b) What is their gross profit (before fines), from making this many boxes of cereal?

c) How often will they need to pay a fine? That is, how many of the 1000 sampled boxes (on average) will weigh less than 20.0 ounces?

d) What is the company’s net profit, i.e., revenue from cereal sales less fines?

General Junkfoods recently hired a new VP for operations, Ismerelda Tempusfugit. She believes that the company has the potential to marginally increase its profits on Chocolate Frosted Sugar Bombs by adjusting the mean setting on the automated box-filling machinery. Thus, for example, by reducing the average amount of cereal in the box to 20.4 ounces, the company will be able to make (and sell) more boxes of cereal. Admittedly, they will need to pay a fine more often. But if the increased revenue from more cereal boxes more than offsets the increased cost of fines, then the venture will be financially worthwhile.

QUESTION 2:

Remember that with the current equipment CANNOT change the standard deviation from 0.2 ounces, but you CAN adjust the mean from its current value of 20.5 ounces.

a) Suppose that General Junkfoods adjusts the process mean to 20.4 ounces. Now what will be the company’s net profit (sales revenue less fines)?

b) To what value should the process mean be set, to maximize the company profit? (HINT: the easiest way to do this may be to set up a spreadsheet that will compute the net profit, given the mean amount of cereal in the box. You can then readily adjust the mean to find the "best" value — the one giving optimum profit.)

Ismerelda Tempusfugit realizes that General Junkfoods cannot at present afford a complete overhaul of the production line. However, it may be possible to afford some renovations to existing equipment that will reduce the variation in the box-filling process. General Junkfoods’ engineering department estimates that it will cost $1,000,000 to reduce the process standard deviation from 0.2 ounces to 0.1 ounces. General Junkfoods VP of Finance, Dewey Cheatham, indicates that this expenditure will be affordable, IF the company can recoup the cost (from increased sales) within two years.

QUESTION 3:

Now suppose the process standard deviation is 0.1 ounces rather than 0.2.

a) Under this scenario, to what value should the process mean be set, to maximize the company profit?

b) How much annual net profit (sales less fines) will the company earn?

c) Should they company spend the $1,000,000 needed to reduce process variation? Explain.