ec41lecture12

ec41lecture12 - Statistics for Economists Lecture 12 Kata...

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Statistics for Economists Lecture 12 Kata Bognar UCLA Central Limit Theorem Approximations Special Continuous Distributions Statistics for Economists Lecture 12 Kata Bognar UCLA November 9, 2010
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Statistics for Economists Lecture 12 Kata Bognar UCLA Central Limit Theorem Approximations Special Continuous Distributions Announcements Midterm solutions are online
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Statistics for Economists Lecture 12 Kata Bognar UCLA Central Limit Theorem Approximations Special Continuous Distributions Last Lecture Normal distributions
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Statistics for Economists Lecture 12 Kata Bognar UCLA Central Limit Theorem Approximations Special Continuous Distributions Today’s Outline 1 Central Limit Theorem 2 Approximations 3 χ 2 and t distributions. 4 Readings: TH, Chapter 3.6 - 3.7, Chapter 3.3: p.124 - 125. Chapter 4.1: p.156 - 158. Suggested readings: WS, Chapter 7.3 W, Chapter 7.3, 6.5 5 Readings for the next class: TH, Chapter 2.5: p.82 - 83; Chapter 3.5: p.139 - 141; Chapter 4.3: p.162 - 164.
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Statistics for Economists Lecture 12 Kata Bognar UCLA Central Limit Theorem Approximations Special Continuous Distributions Distribution of the Sample Mean Suppose X 1 , X 2 ,... X n is a random sample from a distribution with a mean μ and a variance σ 2 . The sample mean is ¯ X = i X i n . the sample mean is a random variable the expected value of the sample mean is E [ ¯ X ] = μ the variance of the sample mean is Var [ ¯ X ] = σ 2 / n Can we say something about the distribution of the sample mean ? Can we say something about the probability that the sampling error is in some range?
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Statistics for Economists Lecture 12 Kata Bognar UCLA Central Limit Theorem Approximations Special Continuous Distributions Sampling from a Normal Distribution Suppose that X is a normal random variable with a mean μ and a variance σ 2 and X 1 , X 2 ,... X n is a random sample from this distribution. Then the sample mean ¯ X has a normal distribution with a mean μ and a variance σ 2 n : ¯ X N ± μ, σ 2 n ² . The random variable W = ¯ X - μ σ/ n has a standard normal distribution : W = ¯ X - μ σ/ n N (0 , 1) .
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Statistics for Economists Lecture 12 Kata Bognar UCLA Central Limit Theorem Approximations Special Continuous Distributions Sampling from a Normal Distribution - Example The weight of a pizza form CPK is a random variable that has a normal distribution with a mean of 16 ounces and a standard deviation of 1 ounce. Suppose you buy 4 pizzas for a party. What is the probability that the mean weight of the 4 pizzas is more than 17 . 1 ounces? Step 1:
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ec41lecture12 - Statistics for Economists Lecture 12 Kata...

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