Introduction to The Central Limit Theorem （介绍了中心极限定理）.pdfVIP

NCSSM Statistics Leadership Institute Notes The Theory of Inference Introduction to The Central Limit Theorem There are a number of important theorems that govern the sampling distribution of Y . Principal among them stands the Central Limit Theorem. A typical presentation of the theorem is given on page 249 in Statistics, The Exploration and Analysis of Data, 3rd, by Devore and Peck (1997), who state it this way: Let Y denote the mean of the observations in a random sample of size n from a population having a mean m and standard deviation s . Denote the mean of the Y distribution by m and the standard deviation of the Y distribution by s . Y Y Then the following rules hold: Rule 1. mY m s Rule 2. sY This rule is approximately correct as long as no more than n 5% of the population is included in the sample. Rule 3. When the population distribution is normal, the sampling distribution of Y is also normal for any sample size n. Rule 4. (Central Limit Theorem) When n is sufficiently large, the sampling distribution of Y is well approximated by a normal curve, even when the population distribution is not itself normal. The key point about the Central Limit Theorem is that it is a theorem about shape. The derivation for the mean and standard deviation of the sampling distribution of sample means (Rules 1 and 2) does not require an assumption of normality. Let us suppose that Y , Y ,..., Y are 1 2 n