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Unformatted text preview: Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Confidence intervals Artin Armagan Sta. 113 Chapter 7 of Devore October 14, 2009 Artin Armagan Confidence intervals Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Table of contents 1 Normal distribution known variance 2 Large sample CI, or CLT to the rescue 3 Small sample normal, thank Guinness 4 Confidence intervals on the spread or variance 5 Confidence bounds 6 Sample size computations Artin Armagan Confidence intervals Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Uncertainty In the last lecture we learned about point estimates using the MLE. We also learned about uncertainty in the context of Bayesian methods and the posterior density. We now study within the likelihood framework how to think of uncertainty. This is the idea of a confidence interval and in statistics lingo it is the frequentist analog of the Bayesian credible interval. Artin Armagan Confidence intervals Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations Confidence interval of the mean If X 1 , ..., Xn iid ∼ No( μ, σ 2 ) with then we know that Z = ¯ X − μ σ/ √ n ∼ No(0 , 1) . This means that Pr ( − 1 . 96 < Z < 1 . 96) = . 95 . Pr − 1 . 96 < ¯ X − μ σ/ √ n < 1 . 96 ! = . 95 . Pr − 1 . 96 σ √ n < ¯ X − μ < 1 . 96 σ √ n ! = . 95 . Pr − 1 . 96 σ √ n − ¯ X < μ < − ¯ X + 1 . 96 σ √ n ! = . 95 . Pr 1 . 96 σ √ n + ¯ X > μ > ¯ X − 1 . 96 σ √ n ! = . 95 . Pr ¯ X − 1 . 96 σ √ n < μ < ¯ X + 1 . 96 σ √ n ! = . 95 . Artin Armagan Confidence intervals Normal distribution known variance Large sample CI, or CLT to the rescue Small sample normal, thank Guinness Confidence intervals on the spread or variance Confidence bounds Sample size computations A random interval Consider the quantity Pr ¯ X − 1 . 96 σ √ n < μ < ¯ X + 1 . 96 σ √ n ! = . 95 , ¯ X is random but μ is not it is fixed. The interpretation of the above equation is as a random interval ℓ = ¯ X − 1 . 96 σ √ n , u = ¯ X + 1 . 96 σ √ n ! . The interval is centered at the sample mean and extends in either direction by 1 . 96 σ √ n ....
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This note was uploaded on 01/17/2010 for the course STA 113 taught by Professor Staff during the Fall '08 term at Duke.
 Fall '08
 Staff
 Normal Distribution, Variance

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