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§ 9. Confidence intervals s. 1. Basic definitions. A confidenceinterval givesanestimatedrangeof valueswhichislikelytoincludeanunknown populationparameter, theestimatedrangebeing calculatedfromagivensetofsampledata. Letbeapointestimateofthepopulation parametercalculatedfromagivensetof sampledata. * Let and 0. || * Theless is themoreaccurate theestimation is.

Confidencecoefficient ( confidencelevel )isa probabilitywithwhichtheinequality takesplace, i. e. Remark. Thestatisticalmethodsdonotallowustosay thattheestimatorsatisfiestheinequality Wecanonlysayaboutprobabilitywithwhichthis inequalityholds. * . || * * | | , . )|(| * P

Theinterval whichcoverstheunknownparameterwith prescribedprobabilityiscalled confidence interval ( CI ). ), ( ** — estimation accuracy. Theconstructionofthe. CI : 1) pointestimatecalculation ; * 2) thechoiceof confidencelevel (0, 95; 0, 995); 3) thecalculationoftheaccuracy.

s. 2. Distributions of the RV, which are often used in statistics. Chi-squared distribution Let. RV areindependentandn. XXX, . . . , , 21 ). 1, 0(~NXi Then. RV 22 2 2 1 2. . . n. XXX iscalledtobedistributedaccordingto thechi-squareddistributionwith k degreesof freedom.

Probabilitydensityfunction : , 0, 0 , 0, )2/(2 1 )( 2/12/ 2/ 2 x xex nxf xn n where 0 1 )(dtetp tp — Gammafunction. Theplotof : )(2 xf

Theexpectedvalue : . )( 2 n. M Thevariance : . 2)( 2 n. D Thequantileofthedistribution , which corresponds thestatisticalsignificance , isa suchvaluethatthefollowinginequalityholds: 2 2 . )( 22 P Remark. Thevaluesofthequantilescanbefoundin specialtables.

Student’s t-distribution (t-distribution) Let), 1, 0(~1 NX RV hasthechi-squareddistribution with k degreesoffreedom. 2 X Thenthe. RV k. X X Tk /2 1 iscalledtobedistributedaccordingtothet- distributionwith k degreesoffreedom.

Probabilitydensityfunction : . 1 )2/( 2/)1( )( 2/)1(2 k T k t kn n tf k Theplotofthe : )(tf k. T Theexpectedvalue : Thevariance : . 0)(k. TM. 2 )( k k TDk

Thequantileofthet-distribution , which corresponds thestatisticalsignificance , isa suchvaluethatthefollowinginequalityholds: t. )|(|t. TPk Remark. Thevaluesofthequantilescanbefoundin specialtables.

s. 3. Confidence Intervals for Unknown Mean and Known Standard Deviation. Let). , (~a. NX Weknow. , Weshouldfindthe. CIforthe a withconfidence level. Let’sfindaccuracy. Thepointestimateforthemeanis. x

Letnxxx, . . . , , 21 isasampleobtainedfromtheobservationsfor the. RV X. Thevalues nxxx, . . . , , 21 change fromsampletosample. Therefore, wecanassumethat ). , (~a. Nxi Besides , thesamplemean isalsoa. RVwhich hasnormaldistribution , and x

1 1 1 ( ) , n n n k k k E x E x a a n n n 2 2 1 1 1 Var( ) , n n n k k k x x x n n i. e. ~ , . x N a n

Since, 2)|(|0 a. XP then 0(| | ) 2. n P x a × Letusdenote. t n × Then , n t and. 2 )(, )(200 tt

Therefore 02 , P x t a x t t n n i. e. withconfidencelevelwecanassertthan. CI , x t n n coversunknownparameter a , andtheaccuracy oftheestimationis. n t

Example. Letwehavesampleofthe. RV : )20, (~a. NX. 21; 10; 20; 34; 25 Find 95% confidenceintervalforthemean. Solution. 25 34 20 10 21 4; 5 x ; 95, 0 ; 475, 0 2 ; 96, 1475, 0)(0 tt. 5, 17 5 20 96, 1× Confidenceinterval : ). 5, 21; 5, 13()5, 174; 5, 174(

s. 4. Confidence. Intervalsfor. Unknown Meanand. Unknown. Standard. Deviation. Let Weknow). , (~a. NX. Weshouldfindthe. CIforthe a withconfidence level. Let’sfindaccuracy. Thepointestimateforthemeanis . x

Let S isastandarderror. Considerthefollowing. RV. / x a T S n Wecanprovethat T hast-distributionwith degreesoffreedom. Let’s find sothat (| | ). P x a 1 n

Letusdividethebothsidesoftheinequalityin bracketson: /n. S , / / x a P S n or. / n. S TP Letusdenote. /n. S t Then. n S t Thevaluecanbedeterminedbythe t-distributiontable. t

i. e. withconfidencelevelwecanassertthan. CITherefore, / x a P t S n or , S S P x t a x t n n ; S S x t n n . n S t coversunknownparameter a , andtheaccuracy oftheestimationis

Example. Inthepreviousexamplefind. CIforthe unknownmean , ifstandarddeviationisunknown. Solution. Let’sfind S : ; 4, 544)1211101)20(1341)25(( 5 1222222 ×××××x 2 2 544, 4 4 528, 4; ns ; 5, 6604, 528 15 52 S. 7, 255, 660 S

Sincetheconfidencelevelis, 95, 0 thenstatisticalsignificance. 05, 01 Theamountofthedegreesoffreedom. 41 n Usingthespecialtable. 78, 2 t Accuracy. 9, 31 5 7, 25 78, 2× Confidenceinterval : ). 9, 35; 9, 27()9, 314; 9, 314(