c h a p t e r 7 The Normal Distribution Introduction Beverage cans are made from a very thin sheet of aluminum, only 1∕80 inch thick. Yet they must withstand pressures of up to 90 pounds per square inch (approximately three times the pressure in an automobile tire). Beverage companies often purchase cans in large shipments. To ensure that can failures are rare, quality control inspectors sample several cans from each shipment and test their strength by placing them in testing machines that apply force until the can fails (is punctured or crushed). The testing process destroys the cans, so the number of cans that can be tested is limited. Assume that a can is considered defective if it fails at a pressure of less than 90 pounds per square inch. The quality control inspectors want the proportion of defective cans to be no more than 0.001, or 1 in 1000. They test 10 cans, with the following results. Can 1 2 3 4 5 6 7 8 9 10 Pressure at Failure 95 96 98 99 99 100 101 101 103 104 Although none of the 10 cans were defective, this is not enough by itself to determine whether the proportion of defective cans is less than 1 in 1000. To make this determination, we must know something about the probability distribution of the pressures at which cans fail. In this chapter, we will study the normal distribution, which is the most important distribution in statistics. In the case study at the end of this chapter, we will show that if the pressures follow a normal distribution, we can estimate the proportion of defective cans. 285
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