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sta 2006 0708_chapter_3

# sta 2006 0708_chapter_3 - STA 2006 2007-2008 Term II...

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STA 2006 2007-2008 Term II Chapter 3 Confidence Intervals for Normal Population (H-T Section 6.4) 1 Known σ Case Suppose X 1 , . . . , X n i.i.d. from the normal distribution with unknown mean μ but known variance σ 2 . Both the method of maximum likelihood and the method of moments suggest using ¯ X n = n i =1 X i /n to estimate μ . Is ¯ X n a ”good” estimator of μ ? ¯ X n is unbiased. That is, E ( ¯ X n ) = μ . ¯ X n is consistent. That is, when n is large, ¯ X n is very close to μ (in terms of probability). This is a direct consequence of the Weak Law of Large Numbers. However, Pr( ¯ X n = μ ) = 0 no matter how large n is because ¯ X n is a continuous random variable. Therefore, the best we can do is to find a (random) interval [ A n , B n ] such that Pr( A n < μ < B n ) is large. First, note that ¯ X n N ( μ, σ 2 /n ) . (1) Therefore, Pr( - z α 2 ¯ X n - μ σ/ n z α 2 ) = 1 - α Here we take α to be small such that 1- α is large. Also z α/ 2 is the standard normal score such that the area over the positive tail is α/ 2 . The following are some useful values 100(1- α )% α/ 2 z α/ 2 90% 0.05 1.645 95% 0.025 1.96 99% 0.005 2.58 1

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Since Pr( ¯ X n - z α 2 σ n μ ¯ X n + z α 2 σ n ) = 1 - α, the 100(1- α )% confidence interval of μ is [ ¯ X n - z α 2 σ n , ¯ X n + z α 2 σ n ] . (2) In reality, there is only one sample. We can only compute a realization of the confidence interval: x n - z α 2 σ n , ¯ x n + z α 2 σ n ] .
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sta 2006 0708_chapter_3 - STA 2006 2007-2008 Term II...

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