UCLA STATS 13  The Central Limit Theorem (10 pages)
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The Central Limit Theorem
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Lecture Notes
 Pages:
 10
 School:
 University of California, Los Angeles
 Course:
 Stats 13  Introduction to Statistical Methods for Life and Health Sciences
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University of California Los Angeles Department of Statistics Statistics 13 Instructor Nicolas Christou The central limit theorem The distribution of the sample proportion The distribution of the sample mean The distribution of the sample proportion First Population proportion p Sample proportion p Very important The sample q proportion p follows the normal distribution with mean p and standard deviation p 1 p where p is the population proportion and n is the n sample size Requirements large n and independent sample values What does the above statement mean 1 We can write s p N p p 1 p n Therefore p p Z q p 1 p n How can we use the above statement Here is an example Suppose the population proportion of Democtrats among UCLA students is 55 Find the probability that the sample proportion of 200 students randomly selected will exceed 60 Important The population proportion p is a parameter it is a fixed number between 0 and 1 It is not random The sample proportion p is not fixed It is random 2 The distribution of the sample mean First Population mean Sample mean X Very important The sample mean X follows the normal distribution with mean and standard deviation n where and are the mean and standard deviation of the population from where the sample of size n is selected Requirements large n and independent sample values What does the above statement mean 3 We can write X N n Therefore Z X n As a result of the previous statement the following is also true and very useful T N n n Therefore Z T n n How can we use the above statement Here is an example A large freight elevator can transport a maximum of 9800 pounds Suppose a load of cargo containing 49 boxes must be transported via the elevator Experience has shown that the weight of boxes of this type of cargo follows a distribution with mean 205 pounds and standard deviation 15 pounds Based on this information what is the probability that all 49 boxes can be safely loaded onto the freight elevator and transported Important
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