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# lec5.1 - 105 Lecture 11 CHAPTER 4 COMMONLY USED...

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105 Lecture 11: CHAPTER 4: COMMONLY USED DISTRIBUTIONS The Central Limit Theorem (Section 4.11, page 289) The Central Limit Theorem is by far the most important result in statistics. Many commonly used statistical methods rely on this theorem for their validity. The Central Limit Theorem says: Mean, Variance and Standard Deviation of the Sample Mean X Given: 1. A population with mean and standard deviation . 2. A random sample of size n from the population: X 1 , X 2 , … X n with sample mean X = (X 1 +X 2 +… X n )/n Then: x [the mean of the distribution of x is equal to the population mean] n x 2 2 [the variance of the distribution of x is equal to (the population variance)/n]

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106 n x [the standard deviation of the distribution of x is equal to the (population standard deviation)/ n ] THE CENTRAL LIMIT THEOREM Given a large random sample of size n from a population with mean and standard deviation , then the sample mean x is approximately normally distributed with mean and standard deviation given by  x n x with x x x z Note: 1. For the central limit theorem to apply, you need a large sample size. n > 30 is usually large enough. 2. If the population from which we sample is normal then x is exactly normally distributed with mean and standard deviation as above for any sample size. Example: Let X denote the number of flaws in a 1 inch length of copper wire. The probability mass function of X is presented in the following table: x P ( X = x ) 0 0.48 1 0.39 2 0.12 3 0.01
107 One hundred wires are sampled from this population. What is the probability that the average number of flaws per wire in this sample is less than 0.5?

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## This note was uploaded on 06/20/2011 for the course STAT 2800 taught by Professor Paula during the Winter '11 term at UOIT.

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lec5.1 - 105 Lecture 11 CHAPTER 4 COMMONLY USED...

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