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# x n isdefinedas x 1n x n i1 i x

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Unformatted text preview: and therefore, the calculated sample means are different. 2 Chapter 7 The sample mean of the random variables X 1 , X 2 , . . . , X n is defined as: X= 1n ∑X n i=1 i X is a linear combination of random variables and, therefore, is also a random variable. X has a probability distribution known as the sampling distribution. The sampling distribution of a sample statistic is the probability distribution of the values it could take over all possible samples of size n drawn from the population. What are the properties of the sampling distribution of X ? First, state the mean: 1n ⎡1 n ⎤ E( X ) = E ⎢ ∑ X i ⎥ = ∑ E(X i ) n i=1 ⎣ n i=1 ⎦ 1 = (n μ) n =μ That is, E( X ) = μ . This says – for a large number of samples (say 1000 samples), each with n observations, the average of the calculated sample means will approach the population mean μ . 3 Chapter 7 Now state the variance: 1n ⎡1 n ⎤ = Var ⎢ ∑ X i ⎥ = 2 ∑ Var(X i ) use independence Var( X ) n i=1 ⎣ n i=1 ⎦ 1 = 2 (n σ 2 ) n =σ n 2 Pr...
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