Interval Estimation for Non-Normal Distributions: Derivation of Confidence Interval, Study notes of Statistics

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chapter 8

Internal Estimation

D Interval Estimator

The random (^) interval ( L (^) , U ) (^) forms an^ interval estimator^ of O U - L B (^) the width (^) of interval coverage probability PC^

L (^ Xi,^ '^

• .^ Xn^ )^ s^ O^ - UCH .^.^.. Xn^ ))^4 Ideally , width (^) should be^ as (^) small as possible coverage probability^ should be^ as^ large as^ possible gcInteNalEstimatimformeansofnormaldi5hibuhh 8 Random (^) Sample IX. (^) ,^ -^.^ - Xml from NCN, 64 with^ known^ variance^6 ' IT is^ a good estimator (^) of n "i÷÷÷i÷÷ ZET (^) Margin of^ error^ T

6T T

rt d

O Normalwithunknom#

Replace

6 with unbiased estimator S =

) #^ E.^ Hi^

- E

) ' looll-HY.anfidenceihterralfvtf-X-tE.mx

Fn , Titta .mx#n

) Derivation (^) : Estimated standard^ error E.s.E.LT/t=fT-xnnca.Enln-yE-nxn-ix-n6An-~Zco. )

xh-xh.it#yo87Interalestima-hbnforvarianaofnorma s

• (^) = #Ed,^ Hi^ - F) 2 Let (^) M =^ In-Dsk^ EE. Hi - Tp ' H =^ Fr^ n^ x'n - i 1004-214. (^) Confidence interval^ for 62 :

Find neck^ , aka such^ that ki - - X't- E. n-i

PL Xc^ k^. )^ = PCX >^ ka^ )^ = (^) I

where X n^ x'n - i

k (^) - =

WE, n - I

< : p (^ Kc MT < ka (^) ) =^ (^ TT -^ 6- < (^) ft ) =^ I^

• L ""I÷÷÷;ma t::I÷¥i÷÷ . )