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The Central Limit Theorem You may be wondering why Normal distributions seem to have a special place among the probability distributions we have studied. Indeed, the Empirical Rule that we looked at several weeks ago came directly from a normal distribution. The reason that the normal distribution occurs so frequently in the real world is because of the Central Limit Theorem. We begin by stating a “business student” version of this theorem. Let X be computed by taking a simple random sample of size n from a population with mean µ and variance σ2. Then for “large n,” X will have an approximate Normal distribution. As is always the case when sampling from an infinite population or from a finite population of size N where N >> n, E[ X ] and Var ( X ) 2 n Since the mean and variance uniquely specifies the normal distribution in question, you have everything you need to know about the distribution of X . In particular, you can compute probabilities about X . For a good normal approximation, “large n” means either (1) n > 1 when sampling from a population which itself has a normal distribution, or (2) n > 30 when sampling from populations with arbitrary distributions. Now, what does the Central Limit Theorem say in general? Whenever you are adding random variables (typically assumed to have an identical distribution) together, like S X1 X 2 X 3 X n 100 the sum S will have an approximate normal distribution (when properly normed) as long as there are not terms in the sum that dominate the sum. In essence, this says that the sum obtained by adding a bunch of numbers of approximately the same size will have an approximate normal distribution. How does this apply to X ? Example 1 Incomes in a community are normally distributed with µ = $30,000 and σ = $4,000. Q1: If we select an income at random, what is the probability that the income exceeds $32,000? Q2: Now suppose we take a random sample of size 4. What is the probability that the average income in the sample exceeds $32, 000? Let X = the average income in the sample of size 4. What is the distribution of X ? 101 We have X X What is the probability we wish to compute? In order to standardize X , we need to subtract its mean and divide by its standard deviation: Z X X X X n Thus X 32,000 30,000 P{ X 32,000} P 4000 n 4 2000 P Z 2000 P{Z 1} 0.1587 Example 2 A company which produces sealing lids considers its production process to be properly adjusted if the average diameter of a lid is 4 inches. The standard deviation of the diameters is 0.012 inches. Someone has suggested that the machine is in need of adjustment, so the foreman has taken a sample of 100 lids and has found that x 4.003 inches 102 Everyone seems to believe that this is really close to 4 inches and production should continue. Should the production facility be shut down to make the adjustment? Assume for a moment that µ = 4 inches; that is, there is no need to make an adjustment. The sample mean we saw was a little larger than this. If, in fact, µ = 4 inches, what is the probability of seeing a sample mean as large as we saw? X 4.003 4.000 P{ X 4.003} P 0.012 n 100 .003 P Z .0012 P{Z 2.5} 0.0082 What do you think? 103