L20posted_rev2

# L20posted_rev2 - C.I. for μ from Normal Population with...

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Unformatted text preview: C.I. for μ from Normal Population with Unknown σ Previously assumed either: • • Now σ is unknown, but sample size not large enough for central limit the- orem. Inference needs some additional assumptions. Assume N ( μ,σ 2 ) population. Interest on parameter μ but σ unknown. 1 Given random sample { X 1 ,...,X n } from N ( μ,σ 2 ) population. Previously, C.I. based on standardized score . Z = Natural to replace σ 2 with an estima- tor. Use S 2 = to get a statistic T = called Studentized score . 2 Write T = The numerator is Z a standard normal. To describe the denominator first de- fine the random variable W = ( n- 1) S 2 σ 2 where, recall S 2 = 1 n n X i =1 ( X i- ¯ X ) 2 3 The numerator of S 2 doesn’t depend on the unknown mean μ because it is subtracted out by X i- ¯ X and when S 2 is divided by σ 2 , the variance σ 2 can- cels out in numerator and denomina- tor. Therefore W is dimensionless ran- dom variable. It can be shown that the distribution of W has a chi-square distribution which is a family that depends only on one parameter, called the degrees of freedom , and given in this case by r = n- 1 . 4 Simulated chi square 10000 times with r = 9 and superimposed pdf (red curve). 5 10 15 20 25 30 35 40 45 50 500 1000 1500 2000 2500 3000 3500 4000 5 Matlab code for simulation and fit . Simulate chi-square on 9 degrees of freedom from nor- mal samples of size 10....
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## This note was uploaded on 01/03/2012 for the course EE 1244 taught by Professor Drera during the Fall '10 term at Conestoga.

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L20posted_rev2 - C.I. for μ from Normal Population with...

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