Question:Case 6.3: American Oil Company

Chad Williams, field geologist for the American Oil Company, settled into his first-class seat on the Sun-Air flight between Los Angeles and Oakland, California. Earlier that afternoon, he had attended a meeting with the design engineering group at the Los Angeles New Product Division. He was now on his way to the home office in Oakland. He was looking forward to the one-hour flight because it would give him a chance to reflect on a problem that surfaced during the meeting. It would also give him a chance to think about the exciting opportunities that lay ahead in Australia.

Chad works with a small group of highly trained people at American Oil who literally walk the earth looking for new sources of oil. They make use of the latest in electronic equipment to take a wide range of measurements from many thousands of feet below the earth's surface. It is one of these electronic machines that is the source of Chad's current problem. Engineers in Los Angeles have designed a sophisticated enhancement that will greatly improve the equipment's ability to detect oil. The enhancement requires 800 capacitors, which must operate within0.500.50plus minus , 0.50microns from the specified standard of 12 microns.

The problem is that the supplier can provide capacitors that operate according to a normal distribution, with a mean of 12 microns and a standard deviation of 1 micron. Thus, Chad knows that not all capacitors will meet the specifications required by the new piece of exploration equipment. This will mean that to have at least 800 usable capacitors, American Oil will have to order more than 800 from the supplier. However, these items are very expensive, so he wants to order as few as possible to meet their needs. At the meeting, the group agreed that they wanted a 98% chance that any order of capacitors would contain the sufficient number of usable items. If the project is to remain on schedule, Chad must place the order by tomorrow. He wants the new equipment ready to go by the time he leaves for an exploration trip in Australia. As he reclined in his seat, sipping a cool lemonade, he wondered whether a basic statistical technique could be used to help determine how many capacitors to order.

• The main question is: What is the probability of any one capacitor having a mean between 11.5 and 12.5 when the population mean is 12 with a standard deviation of 1.
• Conceptually, you need to find the area (probability) under the normal curve between 11.5 and 12.5.
• You can find the area either manually, by first finding the z-score using the formula on p.227 (formula 6.2), and then looking up the area (probability) that the z-score represents by using the standard normal distribution table (p.230 in the text or Appendix D.....or....
• You can find the area using the Excel function Norm.Dist (Formulas > More Functions > Statistical). Remember that when you put "TRUE" in the Cumulative box, Excel returns the probability, or area under the curve, from the left-hand side of the curve up to the "X" that you put in:Normdist.gif
• The final probability that you arrive at will be the probability, or percentage of capacitors that fall between 11.5 and 12.5.
• Remember that the company needs 800 capacitors to operate in the range of 11.5 to 12.5.
• Also remember that the group wants a 98% chance that any order of capacitors would contain a sufficient number of usable items.