Lecture 23-3 - C.I. 2nd copy

Lecture 23-3 - C.I. 2nd copy - 1 EXAMPLES OF CONFIDENCE...

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1 EXAMPLES OF CONFIDENCE INTERVALS Lecture 23 ORIE3500/5500 Summer2011 Li Two sided confidence intervals These are of the form I ( X 1 ,...,X n ) = ( ˆ θ l , ˆ θ u ) , where ˆ θ l and ˆ θ u are two statistics, denoting the lower and upper bounds of the interval. One uses two sided confidence intervals when there is no strong preference to bound the parameter of interest on a particular side, but, a general interest to know where θ is. 1 Examples of Confidence Intervals The crux of the story is that in order to find a confidence interval for any parameter, find a reasonable estimator for the parameter. Modify this esti- mator (rescale or relocate, usually involving the parameter) to get a statistic with a distribution without the parameter and then find the upper and lower bound of the parameter using the quantiles of the distribution. Let us see the examples. Example ( σ is known, CI for μ ) Suppose X 1 ,...,X n are i.i.d. N ( μ,σ 2 ) random vari- ables and we want find a 1 - α level confidence interval for μ . As the first step, we need to construct a statistic. When σ is known, we will use ¯ X - μ σ/ n . This is because it follows standard normal. At the second step, we find the two sided confidence interval of level 1 - α . Write P Φ - 1 ( α/ 2) < ¯ X - μ σ/ n < Φ - 1 (1 - α/ 2) · = 1 - α. (1) 1
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1 EXAMPLES OF CONFIDENCE INTERVALS By solving Φ - 1 ( α/ 2) < ¯ X - μ σ/ n < Φ - 1 (1 - α/ 2), we conclude that the 2-sided confidence interval for μ is ¯ X - σ Φ - 1 (1 - α/ 2) n , ¯ X - σ Φ - 1 ( α/ 2) n · . (2) Denote z α as the 1 - α quantiles of standard normal, i.e.
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This note was uploaded on 12/09/2011 for the course IMSE 0301 taught by Professor Song during the Spring '11 term at HKU.

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Lecture 23-3 - C.I. 2nd copy - 1 EXAMPLES OF CONFIDENCE...

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