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Applied Biotechnology & Bioengineering

Review Article Volume 3 Issue 3

New confidence bounds for the mean of a Gaussian distribution versus the classical confidence bounds

Vincent AR Camara

Research Center for Bayesian Applications Inc, USA

Correspondence: Vincent AR Camara, Research Center for Bayesian Applications Inc, Largo, USA

Received: October 20, 2016 | Published: June 22, 2017

Citation: Camara VAR. New confidence bounds for the mean of a Gaussian distribution versus the classical confidence bounds. J Appl Biotechnol Bioeng. 2017;3(3):351-354. DOI: 10.15406/jabb.2017.03.00069

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Abstract

The aim of the present study is to compare confidence intervals for the mean of a Gaussian distribution. Considering the square error loss function, Approximate Bayesian confidence intervals for the mean of a normal population are derived. Using normal data and SAS software, the obtained approximate Bayesian confidence intervals will then be compared to the ones obtained with the well-known classical method.

Numerical results show that both the classical and the new Approximate Bayesian model perform well and that the Approximate Bayesian model has great coverage accuracy.

Keywords: estimation, loss functions, confidence intervals, coverage accuracy, statistical analysis, mean time between failures

Introduction

Bayesian analysis implies the exploitation of suitable prior information and the choice of a loss function in association with Bayes’ Theorem.1,2 It rests on the notion that a parameter within a model is not merely an unknown quantity but rather behaves as a random variable which follows some distribution. In the area of life testing, it is indeed realistic to assume that a life parameter is stochastically dynamic. This assertion is supported by the fact that the complexity of electronic and structural systems is likely to cause undetected component interactions resulting in an unpredictable fluctuation of life parameters. Recently, Drake.3 gave an excellent account for the use of Bayesian statistics in reliability problems. As he points out “He (Bayesian) realizes that his selection of a prior (distribution) to express his present state of knowledge will necessarily be somewhat arbitrary.4,5,6-10 But he greatly appreciates this opportunity to make his entire assumptive structure clear to the world”. “Why should an engineer not use his engineering judgment and prior knowledge about the parameters in the statistical distribution he has picked? For example, if it is the mean time between failures (MTBF) of an exponential distribution that must be evaluated from some tests, he undoubtedly has some idea of what the value will turn to be‘”. In the present study, we shall consider a classical and useful underlying model.11,12 That is, we shall consider the Normal underlying model characterized by

f(x)= 1 2π σ e 1 2 ( xμ σ ) 2 ;x,μ,σ0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb GaaiikaiaadIhacaGGPaGaeyypa0tddaWcaaGcbaqcLbsacaaIXaaa keaanmaakaaakeaajugibiaaikdacqaHapaCaSqabaqcLbsacqaHdp WCaaGaamyza0WaaWbaaSqabKqaafaajugWaiabgkHiTKqbaoaalaaa jeaqbaqcLbmacaaIXaaajeaqbaqcLbmacaaIYaaaaKqbaoaabmaaje aqbaqcfa4aaSaaaKqaafaajugWaiaadIhacqGHsislcqaH8oqBaKqa afaajugWaiabeo8aZbaaaKqaajaawIcacaGLPaaajuaGdaahaaadbe qaaKqzadGaaGOmaaaaaaqcLbsacaGG7aGaaGjbVlabgkHiTiabg6Hi LkablQNiWjaadIhacqWI6jcCcqGHEisPcaGGSaGaaGjbVlabgkHiTi abg6HiLkablQNiWjabeY7aTjablQNiWjabg6HiLkaacYcacqaHdpWC cqWI7jIzcaaIWaaaaa@7552@  (1)

Once the underlying model is found to be normally or approximately normally distributed, to construct confidence intervals for a Normal population mean, the well-known classical approach uses the following models that rely on the standard Normal and the student-t statistics:

( X ¯ t n1α/2 s n , X ¯ + t n1α/2 s n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqddaqadaGcba qddaqdaaGcbaqcLbsacaWGybaaaiabgkHiTiaadshanmaaBaaajeaq baqcLbmacaWGUbGaeyOeI0IaaGymaiaaysW7cqaHXoqycaGGVaGaaG OmaaWcbeaanmaalaaakeaajugibiaadohaaOqaa0WaaOaaaOqaaKqz GeGaamOBaaWcbeaaaaqcLbsacaGGSaqddaqdaaGcbaqcLbsacaWGyb aaaiabgUcaRiaadshanmaaBaaajeaqbaqcLbmacaWGUbGaeyOeI0Ia aGymaiaaysW7cqaHXoqycaGGVaGaaGOmaaWcbeaanmaalaaakeaaju gibiaadohaaOqaa0WaaOaaaOqaaKqzGeGaamOBaaWcbeaaaaaakiaa wIcacaGLPaaaaaa@5A7F@  (2)

( X ¯ Z α/2 σ n , X ¯ + Z α/2 σ n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqddaqadaGcba qddaqdaaqaaKqzGeGaamiwaaaacqGHsislcaWGAbqddaWgaaqcbaua aKqzadGaeqySdeMaai4laiaaikdaaSqabaqddaWcaaGcbaqcLbsacq aHdpWCaOqaa0WaaOaaaOqaaKqzGeGaamOBaaWcbeaaaaqcLbsacaGG SaqddaqdaaqaaKqzGeGaamiwaaaacqGHRaWkcaWGAbqddaWgaaqcba uaaKqzadGaeqySdeMaai4laiaaikdaaSqabaqddaWcaaGcbaqcLbsa cqaHdpWCaOqaa0WaaOaaaOqaaKqzGeGaamOBaaWcbeaaaaaakiaawI cacaGLPaaaaaa@537D@  (3)

Methodology

In the derivation of our Approximate Bayesian confidence bounds for the mean of s normal distribution, the square error loss function has been used. The square error loss function places a small weight on estimates near the true value and proportionately more weight on extreme deviation from the true value of the parameter. Its popularity is due to its analytical tractability in Bayesian modeling. The square error loss is defined as follows:

L SE ( θ ^ ,θ)= ( θ ^ θ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qddaWgaaqcbauaaKqzadGaam4uaiaadweaaSqabaqcLbsacaGGOaGa fqiUdeNbaKaacaGGSaGaeqiUdeNaaiykaiabg2da90WaaeWaaOqaaK qzGeGafqiUdeNbaKaacqGHsislcqaH4oqCaOGaayjkaiaawMcaa0Wa aWbaaSqabKqaafaajugWaiaaikdaaaaaaa@4CE4@ (4)

Considering the Square Error Loss function, the following Approximate Bayesian confidence bounds for the variance of a Normal distribution13 have been derived:

L σ 2 (SE) = i=1 n ( x i μ) 2 n22ln(α/2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qddaWgaaqcbauaaKqzadGaeq4Wdmxcfa4aaWbaaKqzagqabKqzGeqa aKqzGbGaaGOmaaaajugWaiaacIcacaWGtbGaamyraiaacMcaaSqaba qcLbsacqGH9aqpnmaalaaakeaanmaaqahakeaajugibiaacIcacaWG 4bqddaWgaaqcbauaaKqzadGaamyAaaqcbauabaqcLbsacqGHsislcq aH8oqBcaGGPaqddaahaaWcbeqcbauaaKqzadGaaGOmaaaaaSqaaKqz GeGaamyAaiabg2da9iaaigdaaSqaaKqzGeGaamOBaaGaeyyeIuoaaO qaaKqzGeGaamOBaiabgkHiTiaaikdacqGHsislcaaIYaGaciiBaiaa c6gacaGGOaGaeqySdeMaai4laiaaikdacaGGPaaaaaaa@641B@  (5)

U σ 2 (SE) = i=1 n ( x i μ) 2 n22ln(1α/2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGvb qddaWgaaqcbauaaKqzadGaeq4Wdmxcfa4aaWbaaKqzGiqabKqzahqa aKqzGbGaaGOmaaaajugWaiaacIcacaWGtbGaamyraiaacMcaaSqaba qcLbsacqGH9aqpnmaalaaakeaanmaaqahakeaajugibiaacIcacaWG 4bqcfa4aaSbaaKqaafaajugWaiaadMgaaKqaafqaaKqzGeGaeyOeI0 IaeqiVd0Maaiyka0WaaWbaaSqabKqaafaajugWaiaaikdaaaaaleaa jugibiaadMgacqGH9aqpcaaIXaaaleaajugibiaad6gaaiabggHiLd aakeaajugibiaad6gacqGHsislcaaIYaGaeyOeI0IaaGOmaiGacYga caGGUbGaaiikaiaaigdacqGHsislcqaHXoqycaGGVaGaaGOmaiaacM caaaaaaa@668E@  (6)

Using the equation

σ 2 =E( X 2 ) μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCnmaaCaaaleqajeaqbaqcLbmacaaIYaaaaKqzGeGaeyypa0Jaamyr aiaacIcacaWGybqddaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibi aacMcacqGHsislcqaH8oqBnmaaCaaaleqajeaqbaqcLbmacaaIYaaa aaaa@496A@ (7)

Along with equations (5) and (6), the following Approximate Bayesian confidence bounds for a positive mean of a normal distribution.14 have been easily derived:

U μ(SE) =( i=1 n ( x i x ¯ ) 2 n1 + x ¯ 2 i=1 n ( x i x ¯ ) 2 n22ln(α/2) ) 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqddaahaaGcbe qaaKqzGeGaamyva0WaaSbaaKqaafaajugWaiabeY7aTjaacIcacaWG tbGaamyraiaacMcaaSqabaqcLbsacqGH9aqpnmaabmaakeaanmaala aakeaanmaaqahakeaajugibiaacIcacaWG4bqddaWgaaqcKbay=haa jugWaiaadMgaaKaaGeqaaKqzGeGaeyOeI0IabmiEayaaraGaaiyka0 WaaWbaaOqabKazaa2=baqcLbmacaaIYaaaaaGcbaqcLbsacaWGPbGa eyypa0JaaGymaaGcbaqcLbsacaWGUbaacqGHris5aaGcbaqcLbsaca WGUbGaeyOeI0IaaGymaaaacqGHRaWkceWG4bGbaebanmaaCaaakeqa jqgaG9FaaKqzadGaaGOmaaaajugibiabgkHiT0WaaSaaaOqaa0Waaa bCaOqaaKqzGeGaaiikaiaadIhanmaaBaaajqgaG9FaaKqzadGaamyA aaGcbeaajugibiabgkHiTiqadIhagaqeaiaacMcanmaaCaaakeqajq gaG9FaaKqzadGaaGOmaaaaaOqaaKqzGeGaamyAaiabg2da9iaaigda aOqaaKqzGeGaamOBaaGaeyyeIuoaaOqaaKqzGeGaamOBaiabgkHiTi aaikdacqGHsislcaaIYaGaciiBaiaac6gacaGGOaGaeqySdeMaai4l aiaaikdacaGGPaaaaaWccaGLOaGaayzkaaaaa0WaaWbaaOqabKazaa 2=baqcLbmacaaIWaGaaiOlaiaaiwdaaaaaaa@8809@  (7)

L μ(SE) =( i=1 n ( x i x ¯ ) 2 n1 + x ¯ 2 i=1 n ( x i x ¯ ) 2 n22ln(1α/2) ) 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqddaahaaGcbe qaaKqzGeGaamita0WaaSbaaKqaafaajugWaiabeY7aTjaacIcacaWG tbGaamyraiaacMcaaSqabaqcLbsacqGH9aqpnmaabmaakeaanmaala aakeaanmaaqahakeaajugibiaacIcacaWG4bqddaWgaaqcaasaaKqz adGaamyAaaGcbeaajugibiabgkHiTiqadIhagaqeaiaacMcanmaaCa aakeqajaaibaqcLbmacaaIYaaaaaGcbaqcLbsacaWGPbGaeyypa0Ja aGymaaGcbaqcLbsacaWGUbaacqGHris5aaGcbaqcLbsacaWGUbGaey OeI0IaaGymaaaacqGHRaWkceWG4bGbaebanmaaCaaakeqajaaibaqc LbmacaaIYaaaaKqzGeGaeyOeI0sddaWcaaGcbaqddaaeWbGcbaqcLb sacaGGOaGaamiEa0WaaSbaaKaaGeaajugWaiaadMgaaOqabaqcLbsa cqGHsislceWG4bGbaebacaGGPaqddaahaaGcbeqcaasaaKqzadGaaG OmaaaaaOqaaKqzGeGaamyAaiabg2da9iaaigdaaOqaaKqzGeGaamOB aaGaeyyeIuoaaOqaaKqzGeGaamOBaiabgkHiTiaaikdacqGHsislca aIYaGaciiBaiaac6gacaGGOaGaaGymaiabgkHiTiabeg7aHjaac+ca caaIYaGaaiykaaaaaSGaayjkaiaawMcaaaaanmaaCaaakeqajaaiba qcLbmacaaIWaGaaiOlaiaaiwdaaaaaaa@7FB7@  (8)

Hence, for a normal random variable X with a mean that is smaller or equal to zero, we can infer the following Approximate Bayesian confidence bounds15-17 for the population mean:

U μ(SE) = ( i=1 n ( y i y ¯ ) 2 n1 + y ¯ 2 i=1 n ( y i y ¯ ) 2 n22ln(α/2) ) 0.5 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqddaahaaGcbe qaaKqzGeGaamyvaKqbaoaaBaaajeaqbaqcLbmacqaH8oqBcaGGOaGa am4uaiaadweacaGGPaaajeaqbeaajugibiabg2da9aaanmaabmaaju g4beaanmaalaaajug4beaanmaaqahajug4beaajugibiaacIcacaWG 5bqddaWgaaqcLbyabaqcLbmacaWGPbaajug4beqaaKqzGeGaeyOeI0 IabmyEayaaraGaaiyka0WaaWbaaKqzGhqabKqzagqaaKqzadGaaGOm aaaaaKqzGhqaaKqzGeGaamyAaiabg2da9iaaigdaaKqzGhqaaKqzGe GaamOBaaGaeyyeIuoaaKqzGhqaaKqzGeGaamOBaiabgkHiTiaaigda aaGaey4kaSIabmyEayaaraqddaahaaqcLbEabeqcLbyabaqcLbmaca aIYaaaaKqzGeGaeyOeI0sddaWcaaqcLbEabaqddaaeWbqcLbEabaqc LbsacaGGOaGaamyEa0WaaSbaaKqzagqaaKqzadGaamyAaaqcLbEabe aajugibiabgkHiTiqadMhagaqeaiaacMcanmaaCaaajug4beqajugG beaajugWaiaaikdaaaaajug4beaajugibiaadMgacqGH9aqpcaaIXa aajug4beaajugibiaad6gaaiabggHiLdaajug4beaajugibiaad6ga cqGHsislcaaIYaGaeyOeI0IaaGOmaiGacYgacaGGUbGaaiikaiabeg 7aHjaac+cacaaIYaGaaiykaaaaaOGaayjkaiaawMcaa0WaaWbaaKqz GhqabKqzagqaaKqzadGaaGimaiaac6cacaaI1aaaaKqzGeGaeyOeI0 Iaamyyaaaa@9B77@  (9)

L μ(SE) = ( i=1 n ( y i y ¯ ) 2 n1 + y ¯ 2 i=1 n ( y i y ¯ ) 2 n22ln(1α/2) ) 0.5 a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqddaahaaGcbe qaaKqzGeGaamita0WaaSbaaKqaafaajugWaiabeY7aTjaacIcacaWG tbGaamyraiaacMcaaSqabaqcLbsacqGH9aqpaaqddaqadaqcLbEaba qddaWcaaqcLbEabaqddaaeWbqcLbEabaqcLbsacaGGOaGaamyEa0Wa aSbaaKqzagqaaKqzadGaamyAaaqcLbEabeaajugibiabgkHiTiqadM hagaqeaiaacMcanmaaCaaajug4beqajugGbeaajugWaiaaikdaaaaa jug4beaajugibiaadMgacqGH9aqpcaaIXaaajug4beaajugibiaad6 gaaiabggHiLdaajug4beaajugibiaad6gacqGHsislcaaIXaaaaiab gUcaRiqadMhagaqea0WaaWbaaKqzGhqabKqzagqaaKqzadGaaGOmaa aajugibiabgkHiT0WaaSaaaKqzGhqaa0WaaabCaKqzGhqaaKqzGeGa aiikaiaadMhanmaaBaaajug4beaajugibiaadMgaaKqzGhqabaqcLb sacqGHsislceWG5bGbaebacaGGPaqddaahaaqcLbEabeqcLbyabaqc LbmacaaIYaaaaaqcLbEabaqcLbsacaWGPbGaeyypa0JaaGymaaqcLb EabaqcLbsacaWGUbaacqGHris5aaqcLbEabaqcLbsacaWGUbGaeyOe I0IaaGOmaiabgkHiTiaaikdaciGGSbGaaiOBaiaacIcacaaIXaGaey OeI0IaeqySdeMaai4laiaaikdacaGGPaaaaaGccaGLOaGaayzkaaqd daahaaqcLbEabeqcLbyabaqcLbmacaaIWaGaaiOlaiaaiwdaaaqcLb sacqGHsislcaWGHbaaaa@9C17@  (10)

Where y =x+a and “a” is a constant such that x+a >0

Numerical and results

For the numerical results, we will use samples that have been obtained from normally distributed populations 18-20 (Examples 1, 2, 3, .4, 7) and approximately normal populations (Examples 5, 6). SAS software is used to obtain the normal population mean μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBaaa@3A4C@ and standard deviation σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHdp WCaaa@3A59@  corresponding to each of the Normal and approximately Normal data sets that are given below. The lengths of the classical and Approximate Bayesian confidence intervals are respectively denoted by W C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGxb qddaWgaaqcbauaaKqzadGaam4qaaWcbeaaaaa@3BEB@  and W SE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGxb qddaWgaaqcbauaaKqzadGaam4uaiaadweaaSqabaaaaa@3CC5@ .

Example 1: Data obtained from Prem S Mann21 24, 28, 22, 25, 24, 22, 29, 26, 25, 28, 19, 29. Normal population distribution obtained with SAS:

N(μ=25.083,σ=3.1176) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaaiikaiabeY7aTjabg2da9iaaikdacaaI1aGaaiOlaiaaicdacaaI 4aGaaG4maiaacYcacqaHdpWCcqGH9aqpcaaIZaGaaiOlaiaaigdaca aIXaGaaG4naiaaiAdacaGGPaaaaa@49C3@

x ¯ =25.08333 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b GbaebacqGH9aqpcaqGYaGaaeynaiaab6cacaqGWaGaaeioaiaaboda caqGZaGaae4maaaa@405F@

The corresponding (Table 1) sample mean and sample variance is

s 2 =9.719696 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qddaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiabg2da9iaaiMda caGGUaGaaG4naiaaigdacaaI5aGaaGOnaiaaiMdacaaI2aaaaa@4388@

C.L.%

Approx. Bayesian Bounds (SE)

Classical Bounds

WC WSE

80

25.0683-25.1311

23.85665-26.31001

39.87

90

25.0661-25.1437

23.46696-26.69971

41.66

95

25.0650-25.1543

23.10246-27.06420

44.36

99

25.0641-25.1734

22.28798-27.87869

51.15

Table 1 Classical and Approximate Bayesian confidence intervals for the population mean corresponding to the first example of data set

Example 2: Data obtained from Prem S Mann21 13, 11, 9, 12, 8, 10, 5, 10, 9, 12, 13. Normal population distribution obtained with SAS:

N(μ=10.182,σ=2.4008) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaaiikaiabeY7aTjabg2da9iaaigdacaaIWaGaaiOlaiaaigdacaaI 4aGaaGOmaiaacYcacqaHdpWCcqGH9aqpcaaIYaGaaiOlaiaaisdaca aIWaGaaGimaiaaiIdacaGGPaaaaa@49B9@

x ¯ =10.181812 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b GbaebacqGH9aqpcaqGXaGaaeimaiaab6cacaqGXaGaaeioaiaabgda caqG4aGaaeymaiaabkdaaaa@4110@

The corresponding (Table 2) sample mean and sample variance is

s 2 =5.763636 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qddaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiabg2da9iaaiwda caGGUaGaaG4naiaaiAdacaaIZaGaaGOnaiaaiodacaaI2aaaaa@437D@

C.L.%

Approximate Bayesian Bounds (SE)

Classical Bounds

WC WSE

0

10.1575-10.2565

9.18869-11.17495

20.06

90

10.1538-10.2756

8.87019-11.49344

21.54

95

10.1520-10.2914

8.56907-11.79457

23.14

99

10.1506-10.3194

7.88792-12.47572

27.18

Table 2 Classical and Approximate Bayesian confidence intervals for the population mean corresponding to the second example of data set

Example 3: Data obtained from Prem S Mann21 16, 14, 11, 19, 14, 17, 13, 16, 17, 18, 19, 12. Normal population distribution obtained with SAS:

N(μ=15.5,σ=2.6799) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaaiikaiabeY7aTjabg2da9iaaigdacaaI1aGaaiOlaiaaiwdacaGG SaGaeq4WdmNaeyypa0JaaGOmaiaac6cacaaI2aGaaG4naiaaiMdaca aI5aGaaiykaaaa@4857@

The corresponding (Table 3) sample mean and sample variance is

x ¯ =15.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b GbaebacqGH9aqpcaqGXaGaaeynaiaab6cacaqG1aaaaa@3D86@

s 2 =7.181818 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qddaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiabg2da9iaaiEda caGGUaGaaGymaiaaiIdacaaIXaGaaGioaiaaigdacaaI4aaaaa@437B@

C.L.%

Approx. Bayesian Bounds (SE)

Classical Bounds

WC WSE

80

15.4820-15.5570

14.44556-16.55440

28.12

90

15.4794-15.5721

14.11058-16.88942

29.98

95

15.4781-15.5847

13.79727-17.20273

31.95

99

15.4770-15.6075

13.09714-17.90286

36.83

Table 3 Classical and Approximate Bayesian confidence intervals for the population mean corresponding to the third example of data set

Example 4: Data obtained from Prem S Mann21 27, 31, 25, 33, 21, 35, 30, 26, 25,31.33.30, 28. Normal population distribution obtained with SAS:

N(μ=28.846,σ=3.9549) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaaiikaiabeY7aTjabg2da9iaaikdacaaI4aGaaiOlaiaaiIdacaaI 0aGaaGOnaiaacYcacqaHdpWCcqGH9aqpcaaIZaGaaiOlaiaaiMdaca aI1aGaaGinaiaaiMdacaGGPaaaaa@49D9@

The corresponding (Table 4) sample mean and sample variance is

s 2 =15.641025 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qddaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiabg2da9iaaigda caaI1aGaaiOlaiaaiAdacaaI0aGaaGymaiaaicdacaaIYaGaaGynaa aa@442B@

x ¯ =28.846153 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b GbaebacqGH9aqpcaqGYaGaaeioaiaab6cacaqG4aGaaeinaiaabAda caqGXaGaaeynaiaabodaaaa@411F@

C.L.%

Approximate Bayesian Bounds (SE)

Classical Bounds

WC WSE

80

28.8270-28.9087

27.35878-30.33353

36.41

90

28.8242-28.9256

26.89151-30.80080

38.55

95

28.8228-28.9400

26.45604-31.2362

40.79

99

28.8217-28.9663

25.49517-32.19714

46.35

Table 4 Classical and Approximate Bayesian confidence intervals for the population mean corresponding to the fourth example of data set

Example 5: Data obtained from James T et al.22 52, 33, 42, 44, 41, 50, 44, 51, 45, 38,37,40,44, 50, 43. Normal population distribution obtained with SAS:

N(μ=43.6,σ=5.4746) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaaiikaiabeY7aTjabg2da9iaaisdacaaIZaGaaiOlaiaaiAdacaGG SaGaeq4WdmNaeyypa0JaaGynaiaac6cacaaI0aGaaG4naiaaisdaca aI2aGaaiykaaaa@4852@

The corresponding (Table 5) sample mean and sample variance is

x ¯ =43.6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b GbaebacqGH9aqpcaqG0aGaae4maiaab6cacaqG2aaaaa@3D88@

s 2 =29.971428 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qddaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiabg2da9iaaikda caaI5aGaaiOlaiaaiMdacaaI3aGaaGymaiaaisdacaaIYaGaaGioaa aa@443D@

C.L.%

Approximate Bayesian Bounds (SE)

Classical Bounds

WC WSE

80

43.5794-43.6703

41.69879-45.50121

41.83

90

43.5764-43.6902

41.11076-46,08924

43.75

95

43.5749-43.7074

40.56796-46.63204

63.3

99

43.5738-43.7395

39.39189-47.80811

50.79

Table 5 Classical and Approximate Bayesian confidence intervals for the population mean corresponding to the fifth example of data set

Example 6: Data obtained from James T et al.22 52, 43, 47, 56, 62, 53, 61, 50, 56, 52, 53, 60, 50, 48, 60, 55. Normal population distribution obtained with SAS:

N(μ=53.625,σ=5.4145) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaaiikaiabeY7aTjabg2da9iaaiwdacaaIZaGaaiOlaiaaiAdacaaI YaGaaGynaiaacYcacqaHdpWCcqGH9aqpcaaI1aGaaiOlaiaaisdaca aIXaGaaGinaiaaiwdacaGGPaaaaa@49C7@

The corresponding (Table 6) sample mean and sample variance is

x ¯ =53.625 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b GbaebacqGH9aqpcaqG1aGaae4maiaab6cacaqG2aGaaeOmaiaabwda aaa@3EF6@

s 2 =29.316666 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qddaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiabg2da9iaaikda caaI5aGaaiOlaiaaiodacaaIXaGaaGOnaiaaiAdacaaI2aGaaGOnaa aa@443A@

C.L.%

Approximate Bayesian Bounds (SE)

Classical Bounds

WC WSE

80

53.6098-53.6779

51.80979-55.44021

53.31

90

53.6076-53.6932

51.25210-55.99790

55.44

95

53.6065-53.7064

50.74043-56.50957

57.75

99

53.6056-53.7315

49.63588-57.61412

63.37

Table 6 Classical and Approximate Bayesian confidence intervals for the population mean corresponding to the sixth example of data set

Example 7: The following observations have been obtained from the collection of SAS data sets.8 50, 65, 100, 45, 111, 32, 45, 28, 60, 66, 114, 134, 150, 120, 77, 108, 112, 113, 80, 77, 69, 91, 116, 122, 37, 51, 53, 131, 49, 69, 66, 46, 131, 103, 84, 78. Normal population distribution obtained with S

N(μ=82.861,σ=33.226) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGob GaaiikaiabeY7aTjabg2da9iaaiIdacaaIYaGaaiOlaiaaiIdacaaI 2aGaaGymaiaacYcacqaHdpWCcqGH9aqpcaaIZaGaaG4maiaac6caca aIYaGaaGOmaiaaiAdacaGGPaaaaa@49C8@

The corresponding (Table 7) sample mean and sample variance is

x ¯ =82.8611 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b GbaebacqGH9aqpcaqG4aGaaeOmaiaab6cacaqG4aGaaeOnaiaabgda caqGXaaaaa@3FAE@

s 2 =1103.951587 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGWj0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGZb qddaahaaWcbeqcbauaaKqzadGaaGOmaaaajugibiabg2da9iaaigda caaIXaGaaGimaiaaiodacaGGUaGaaGyoaiaaiwdacaaIXaGaaGynai aaiIdacaaI3aaaaa@45AF@

C.L.%

Approximate Bayesian Bounds (SE)

Classical Bounds

WC WSE

80

82.7072-83.4808

75.6261-90.0959

18.7

90

82.6856-83.6884

73.5052-92.2168

18.66

95

82.6751-83.8815

71.6196-94.1024

18.64

99

82.6669-84.2823

67.7793-97.9427

18.67

Table 7 Classical and Approximate Bayesian confidence intervals for the population mean corresponding to the seventh example of data set

All the above tables show that the obtained Approximate Bayesian confidence intervals contain the population mean and are strictly included in their classical counterparts; also, the widths of the classical confidence intervals are more than twenty times greater than the ones corresponding to their Approximate Bayesian counterparts.

Summary and conclusion

  1. The classical and our Approximate Bayesian models perform well. The Approximate Bayesian model has great coverage accuracy and performs better.
  2. The classical method used to constructing confidence intervals for the mean of a Normal population does not always yield the best coverage accuracy. In fact, the above Approximate Bayesian models perform better than their classical counterparts.
  3. Contrary to the classical method that uses the standard Normal and the student-t statistics, the new Approximate Bayesian approach and confidence bounds rely only on the observations that are under study.
  4. With the new Approximate Bayesian approach, Approximate Bayesian confidence intervals for a Normal or approximately Normal population mean are easily obtained for any level of significance.
  5. Bayesian Analysis contributes to reinforcing well-known statistical theories such as the Estimation and Decision-Making theories.

Acknowledgements

None.

Conflict of interest

The author declares no conflict of interest.

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