07. Central limit theorem - proof and examples

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

### Page1 / 12

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online