Don Campbell and his younger brother, Edward, purchased Camp-bell Brewery from their father in 1983. The brewery makes and bottles beer under two labels and distributes it throughout the Southwest. Since purchasing the brewery, Don has been instru-mental in modernizing operations.
One of the latest acquisitions is a filling machine that can be adjusted to fill at any average fill level desired. Because the bottles and cans filled by the brewery are exclusively the 12-ounce size, when they received the machine, Don set the fill level to 12 ounces and left it that way. According to the manufacturer’s specifications, the machine will fill bottles or cans around the average, with a standard deviation of 0.15 ounce.
Don just returned from a brewery convention at which he attended a panel discussion related to problems with filling machines. One brewery representative discussed a problem her company had. It failed to learn that its machine’s average fill went out of adjustment until several months later, when its cost accounting department reported some problems with beer production in bulk not matching output in bottles and cans. It turns out that the machine’s average fill had increased from 12 ounces to 12.07 ounces. With large volumes of production, this deviation meant a substantial loss in profits.
Another brewery reported the same type of problem, but in the opposite direction. Its machine began filling bottles with slightly less than 12 ounces on the average. Although the consumers could not detect the shortage in a given bottle, the state and federal agencies responsible for checking the accuracy of packaged products discovered the problem in their testing and substantially fined the brewery for the underfill.
These problems were a surprise to Don Campbell. He had not considered the possibility that the machine might go out of adjustment and pose these types of problems. In fact, he became very concerned because the problems of losing profits and potentially being fined by the government were ones that he wished to avoid, if possible. After the convention, Don and Ed decided to hire a consulting firm with expertise in these matters to assist them in set-ting up a procedure for monitoring the performance of the filling machine.
The consultant suggested that they set up a sampling plan in which once a month, they would sample some number of bottles and measure their volumes precisely. If the average of the sample deviated too much from 12 ounces, they would shut the machine down and make the necessary adjustments. Otherwise, they would let the filling process continue. The consultant identified two types of problems that could occur from this sort of sampling plan:
1. They might incorrectly decide to adjust the machine when it was not really necessary to do so.
2. They might incorrectly decide to allow the filling process to continue when, in fact, the true average had deviated from 12 ounces.
After carefully considering what the consultant told them, Don indicated that he wanted no more than a 0.02 chance of the first problem occurring because of the costs involved. He also decided that if the true average fill had slipped to 11.99 ounces, he wanted no more than a 0.05 chance of not detecting this with his sampling plan. He wanted to avoid problems with state and federal agencies. Finally, if the true average fill had actually risen to 12.007 ounces, he wanted to be able to detect this 98% of the time with his sampling plan. Thus, he wanted to avoid the lost profits that would result from such a problem.
In addition, Don needs to determine how large a sample size is necessary to meet his requirements.