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Basic Statistics 03 Sampling & Central Limit Theorem Econometric Analysis 1 Sampling & C.L.T. 1. The distribution of the sample mean 2. The Central Limit Theorem 3. The application of CLT Econometric Analysis 2 1. Distribution of the Sample Means The sampling distribution of the sample mean is a probability distribution consisting of all possible sample means of a given sample size selected from a population. nµ 1 1 =µ E ( X ) = E ∑ X i = ∑ E( X i ) = n n n 1 1 Var ( X ) = Var ∑ X i = 2 n n *in the case of finite population, nσ 2 σ 2 ∑ Var ( X i ) = n 2 = n Var ( X ) = Econometric Analysis σ2 N −n n ⋅ N −1 3 Population Distribution µ Econometric Analysis Xi 4 Sampling Dist. of the Sample Means n=N 1<n<N n=1 µ Econometric Analysis Xi 5 2. Central Limit Theorem For any population with a mean µ and a variance σ2, the sampling distribution of the means of all possible samples of size n will be approximately normally distributed, with larger sample size n. The mean of the sampling distribution equal to µ and the variance equal to σ2/n. Econometric Analysis 6 CLT The Population Distribution Uniform Distribution 1600 1400 1200 1000 800 600 μ=0.5003 σ2=0.0831 N=250,000 400 200 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Econometric Analysis 0.7 0.8 0.9 1.0 7 CLT The Sample Distribution of Sample mean The Normal Distribution 700 600 500 400 300 E(X) = 0.5002 Var(X) = 0.0028 n = 30 200 (σ2/30 = 0.0831/30) sample size = 15,000 100 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Econometric Analysis 0.7 0.8 0.9 1.0 8 CLT 0 the case of X~N(µ, σ2) If a population follows the normal distribution, the sampling distribution of the sample mean will also follow the normal distribution for any sample size. To determine the probability a sample mean falls within a particular region, use: X~N ( µ , σ2 n ) X −µ z= ~N (0,1) σ n Econometric Analysis 9 CLT 1 the case of X~ non-normal dist.(µ, σ2) If the population isn’t normally distributed (with known σ2) and sample size is large, the sample means will follow the normal distribution. (See the above figure.) d X → N (µ , σ2 n ) X −µ d → N (0,1) σ n Econometric Analysis 10 CLT 2 the case of X~N(µ, unknown σ2) If the population follows the normal distribution but σ2 is unknown, the sample means will follow the t distribution. But with larger sample size (at least n >30), the sample means will follow the normal distribution. X −µ t= ~t (d . f .) s n X −µ d t= → N (0,1) s n Econometric Analysis 11 CLT 3 the case of X~ non-N (µ, unknown σ2) If the population isn’t normally distributed with unknown σ2 and sample size is large, the sample means will follow the t distribution. (with larger sample size, the normal distribution) X −µ d → N (0,1) s n Econometric Analysis 12