Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 08 Large sample confidence intervals Moo K. Chung [email protected] September 27, 2004 1. The sample size is inversely related to the width of confidence interval. Example 7.4. 2. Central Limit Theorem. Let X 1 , · · · , X n be a random sample with mean μ and variance σ 2 . For sufficiently large n , Z = ¯ X - μ σ/ n N (0 , 1) . 3. Let X 1 , · · · , X n be a random sample with mean μ . For sufficiently large n , Z = ¯ X - μ S/ n N (0 , 1) where S is the sample standard deviation. If n is sufficiently large, approximate 100(1 - α )% confidence interval for μ is ¯ x ± z α/ 2 s n , where s is the sample standard deviation. 4. General large sample confidence interval. Sup- pose ˆ θ is an unbiased estimator of some parame- ter θ , Then 100(1 - α )% confidence interval is ˆ θ + z α/ 2 p V ˆ θ. In many applications, V ˆ θ is a function of θ which makes computation of CI complicated. In this sit-
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