07. Central limit theorem - proof and examples

07. Central limit theorem - proof and examples - University...

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University of California, Los Angeles Department of Statistics Statistics 100B Instructor: Nicolas Christou The Central Limit Theorem Suppose that a sample of size n is selected from a population that has mean μ and standard deviation σ . Let X 1 ,X 2 , ··· ,X n be the n observations that are independent and identically distributed (i.i.d.). Define now the sample mean and the total of these n observations as follows: ¯ X = n i =1 X i n T = n X i =1 X i The central limit theorem states that the sample mean ¯ X follows approximately the normal distribution with mean μ and standard deviation σ n , where μ and σ are the mean and stan- dard deviation of the population from where the sample was selected. The sample size n has to be large (usually n 30) if the population from where the sample is taken is nonnormal. If the population follows the normal distribution then the sample size n can be either small or large. To summarize: ¯ X N ( μ, σ n ). To transform ¯ X into z we use: z = ¯ x - μ σ n Example: Let X be a random variable with μ = 10 and σ = 4. A sample of size 100 is taken from this population. Find the probability that the sample mean of these 100 observations is less than 9. We write P ( ¯ X < 9) = P ( z < 9 - 10 4 100 ) = P ( z < - 2 . 5) = 0 . 0062 (from the standard normal probabilities table). Similarly the central limit theorem states that sum T follows approximately the normal distribution, T N ( nμ, ), where μ and σ are the mean and standard deviation of the population from where the sample was selected. To transform T into z we use: z = T - Example: Let X be a random variable with μ = 10 and σ = 4. A sample of size 100 is taken from this population. Find the probability that the sum of these 100 observations is less than 900. We write P ( T < 900) = P ( z < 900 - 100(10) 100(4) ) = P ( z < - 2 . 5) = 0 . 0062 (from the standard normal probabilities table). Below you can find some applications of the central limit theorem. 1
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EXAMPLE 1 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 μ = 205 pounds and standard deviation σ = 15 pounds. Based on this information, what is the probability that all 49 boxes can be safely loaded onto the freight elevator and transported? EXAMPLE 2 From past experience, it is known that the number of tickets purchased by a student standing in line at the ticket window for the football match of UCLA against USC follows a distribution that has mean μ = 2 . 4 and standard deviation σ = 2 . 0. Suppose that few hours before the start of one of these matches there are 100 eager students standing in line to purchase tickets.
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07. Central limit theorem - proof and examples - University...

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