# Lecture23 - ECO220Y Lecture 23 Estimation of when 2 is...

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ECO220Y Lecture 23 CO220 ectu e 23 Estimation of μ when σ 2 is unknown Migiwa Tanaka Reading: 8.4 (pp. 281 – 285), and 12.1 (pp.381 – 383, 386 – 391) 1

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utline Outline Introduction: Estimation of μ when σ 2 is unknown Student t Distribution Application – Estimation of weekly sales 2
stimation of μ when unknown Estimation of μ when σ 2 is unknown Lecture 21 & 22: Estimation of mean of X, μ when σ 2 is known: Under a certain condition, 2 X 1- α CI estimator of μ:   ) 1 , 0 ( ~ , ~ N N X n n z X This lecture: Estimation of mean of X, μ when σ 2 is n 2 / unknown . Since σ is unknown, need to estimate it. A natural candidate estimator: sample standard deviation, s . X No! The complication is that 3 ? ) 1 , 0 ( N n s s is r.v. while σ is constant.

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sing s as an estimator of Using s as an estimator of X is r.v. with mean μ and variance 2  , ~ 2 N X n   , ~ 2 N X n X Z X Y / n / 2 n s n ) 1 , 0 ( N Z   where 1 X X s i i ? ~ 1 Y n 4
sing s as an estimator of Using s as an estimator of X ) 1 , 0 ( ~ / N n s It was found by British Statistician, William S. Gosset, who worked for a brewery in Ireland. Moreover, he found following: If X is normally distributed , called “t statistics” follows Student t istribution with egree of freedom n s X / distribution with degree of freedom, ν . ) ( ~ t X t 5 / n s

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tudent t Distribution (Recap Lecture16) Student t Distribution (Recap, Lecture16) Student t density function is give by: 2 ) 1 ( 2 ! 2 1 y 1 !
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Lecture23 - ECO220Y Lecture 23 Estimation of when 2 is...

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