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5.4 Sampling Distributions and the Central Limit Theorem • Key Concepts: – How to find sampling distributions and verify their properties – The Central Limit Theorem – Using the Central Limit Theorem to find probabilities 5.4 Sampling Distributions and the Central Limit Theorem • What is a sampling distribution? – The probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. • Examples of sample statistics: 2 X , s, s , and p 5.4 Sampling Distributions and the Central Limit Theorem • Properties of sampling distributions of sample means: 1. The mean of the sample means, X , is equal to the population mean : X 2. The standard deviation of the sample means, X , is proportional to σ : X • Note: X n (valid when n ≤ 0.05N) is known as the standard error of the mean. 5.4 Sampling Distributions and the Central Limit Theorem • According to Forbes Magazine, the top five wealthiest women in the world in 2010 were: Christy Walton: Alice Walton: Liliane Bettencourt: Birgit Rausing: Savitri Jindal: $22.5 billion $20.6 billion $20.0 billion $13.0 billion $12.2 billion 1. Let X = wealth (in billions of dollars). Find the mean and standard deviation of X. 5.4 Sampling Distributions and the Central Limit Theorem 2. List all possible samples of size 2 from this population of 5 and list the sample mean for each sample. 3. Find the mean and standard deviation of the sample means in column three. Sample Wealth Sample Mean {C, A } 22.5, 20.6 21.55 { C,L } 22.5, 20.0 21.25 { C,B } 22.5, 13.0 17.75 { C,S } 22.5,12.2 17.35 { A,L } 20.6, 20.0 20.3 { A,B} 20.6, 13.0 16.8 { A,S } 20.6, 12.2 16.4 { L,B } 20.0, 13.0 16.5 { L,S } 20.0, 12.2 16.1 { B,S } 13.0, 12.2 12.6 5.4 Sampling Distributions and the Central Limit Theorem • Where does the Central Limit Theorem fit in? – We use the CLT to make a statement about the shape of the distribution of the sample means (see p. 263) • If samples of size 30 or more are drawn from any population with mean µ and standard deviation σ, then the sampling distribution of the sample means will be approximately normal. • If the population itself is normally distributed, then the sampling distribution of the sample means is normally distributed for any sample size n. 5.4 Sampling Distributions and the Central Limit Theorem • Practice using the Central Limit Theorem #10 p. 270 (Annual Snowfall) #24 p. 271 (Canned Vegetables) #30 p. 272 (Gas Prices: California) #36 p. 272 (Make a Decision)