REAL STATISTICS REAL DECISIONS Putting it all together 292 CHAPTER 5 Normal Probability Distributions You work for a pharmaceuticals company as a statistical process analyst. Your job is to analyze processes and make sure they are in statistical control. In one process, a machine is supposed to add 9.8 milligrams of a compound to a mixture in a vial. (Assume this process can be approximated by a normal distribution with a standard deviation of 0.05.) The acceptable range of amounts of the compound added is 9.65 milligrams to 9.95 milligrams, inclusive. Because of an error with the release valve, the setting on the machine “shifts” from 9.8 milligrams. To check that the machine is adding the correct amount of the compound into the vials, you select at random three samples of five vials and find the mean amount of the compound added for each sample. A coworker asks why you take 3 samples of size 5 and find the mean instead of randomly choosing and measuring the amounts in 15 vials individually to check the machine’s settings. (Note: Both samples are chosen without replacement.) EXERCISES 1. Sampling Individuals Assume the machine shifts and the distribution of the amount of the compound added now has a mean of 9.96 milligrams and a standard deviation of 0.05 milligram. You select one vial and determine how much of the compound was added. (a) What is the probability that you select a vial that is within the acceptable range (in other words, you do not detect that the machine has shifted)? (See figure.) (b) You randomly select 15 vials. What is the probability that you select at least one vial that is within the acceptable range? 2. Sampling Groups of Five Assume the machine shifts and is filling the vials with a mean amount of 9.96 milligrams and a standard deviation of 0.05 milligram. You select five vials and find the mean amount of compound added. (a) What is the probability that you select a sample of five vials that has a mean that is within the acceptable range? (See figure.) (b) You randomly select three samples of five vials. What is the probability that you select at least one sample of five vials that has a mean that is within the acceptable range? (c) Which is more sensitive to a shift of parameters—an individual random selection or a randomly selected sample mean? 3. Writing an Explanation Write a paragraph to your coworker explaining why you take 3 samples of size 5 and find the mean of each sample instead of randomly choosing and measuring the amounts in 15 vials individually to check the machine’s setting. Mean = 9.8 Mean = 9.96 9.7 9.8 9.9 10.0 10.1 Masses (in milligrams) Upper limit of acceptable range x Original distribution of individual vials Distribution when machine shifts FIGURE FOR EXERCISE 1 Mean = 9.96 Upper limit of acceptable range Distribution when machine shifts Original distribution of sample means, n = 5 Mean = 9.8 9.7 9.8 9.9 10.0 10.1 Masses (in milligrams) x FIGURE FOR EXERCISE 2
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