Research Article Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean

Size: px

Start display at page:

Download "Research Article Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean"

Elinor Wilkins
5 years ago
Views:

1 Probability ad Statistics Volume 2013, Article ID , 11 pages Research Article Cofidece Itervals for the Coefficiet of Variatio i a Normal Distributio with a Kow Populatio Mea Wararit Paichkitkosolkul Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techology, Thammasat Uiversity, Phathum Thai 12120, Thailad Correspodece should be addressed to Wararit Paichkitkosolkul; wararit@mathstat.sci.tu.ac.th Received 23 July 2013; Accepted 25 September 2013 Academic Editor: Shei-chug Chow Copyright 2013 Wararit Paichkitkosolkul. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. This paper presets three cofidece itervals for the coefficiet of variatio i a ormal distributio with a kow populatio mea. Oe of the proposed cofidece itervals is based o the ormal approximatio. The other proposed cofidece itervals are the shortest-legth cofidece iterval ad the equal-tailed cofidece iterval. A Mote Carlo simulatio study was coducted to compare the performace of the proposed cofidece itervals with the existig cofidece itervals. Simulatio results have show that all three proposed cofidece itervals perform well i terms of coverage probability ad expected legth. 1. Itroductio The coefficiet of variatio of a distributio is a dimesioless umber that quatifies the degree of variability relative to the mea[1]. It is a statistical measure for comparig the dispersio of several variables obtaied by differet uits. The populatio coefficiet of variatio is defied as a ratio of the populatio stadard deviatio (σ) to the populatio mea (μ) give by κ=σ/μ. The typical sample estimate of κ is give as κ = S X, (1) where S is the sample stadard deviatio, the square root of the ubiased estimator of populatio variace, ad X is the sample mea. The coefficiet of variatio has bee widely used i may areas such as sciece, medicie, egieerig, ecoomics, ad others. For example, the coefficiet of variatio has also bee employed by Ah [2] to aalyze the ucertaity of fault trees. Gog ad Li [3] assessed the stregth of ceramics by usig the coefficiet of variatio. Faber ad Kor [4] applied the coefficiet of variatio as a way of icludig a measureofvariatioithemeasyapticresposeofthe cetral ervous system. The coefficiet of variatio has also bee used to assess the homogeeity of boe test samples to help determie the effect of exteral treatmets o the properties of boes [5]. Billigs et al. [6] used the coefficiet of variatio to study the impact of socioecoomic status ohospitaluseinewyorkcity.ifiaceadactuarial sciece, the coefficiet of variatio ca be used as a measure of relative risk ad a test of the equality of the coefficiets of variatio for two stocks [7]. Furthermore, Pye et al. [8] studied the variability of the competitive performace of Olympicswimmersbyusigthecoefficietofvariatio. Although the poit estimator of the populatio coefficiet of variatio show i (1)cabeausefulstatisticalmeasure,its cofidece iterval is more useful tha the poit estimator. A cofidece iterval provides much more iformatio about the populatio characteristic of iterest tha does a poit estimate (e.g., Smithso [9], Thompso [10], ad Steiger [11]). There are several approaches available for costructig the cofidece iterval for κ.[12] proposed a cofidece iterval for κ based o the chi-square distributio; this cofidece iterval works well whe κ < 0.33 [13 17]. Later, Vagel [18] proposed a ew cofidece iterval for κ, which is called a modified s cofidece iterval. His cofidece iterval is based o a aalysis of the distributio of a class of approximate pivotal quatities for the ormal coefficiet of variatio. I additio, modified s cofidece iterval is closely related to s cofidece

2 2 Probability ad Statistics itervalbutitisusuallymoreaccurateadearlyexact uder ormality. Paichkitkosolkul [19] modified s cofidece iterval by replacig the sample coefficiet of variatio with the maximum likelihood estimator for a ormal distributio. Sharma ad Krisha [20] itroduced the asymptotic distributio ad cofidece iterval of the reciprocal of the coefficiet of variatio which does ot require ay assumptios about the populatio distributio to be made. [21] discussed the approximate distributio of κ ad proposed the approximate cofidece iterval for κ ithecaseofaormaldistributio.theperformaceofmay cofidece itervals for κ obtaied by s, s, ad Sharma-Krisha s methods was compared uder the same simulatio coditios by Ng [22]. Mahmoudvad ad Hassai [23] proposed a approximately ubiased estimator for κ i a ormal distributio ad also used this estimator for costructig two approximate cofidece itervals for the coefficiet of variatio. The cofidece itervals for κ i ormal ad logormal were proposed by Koopmas et al. [24] ad Verrill [25]. Butao ad Niwitpog [26] also itroduced a iterval estimatig the differece of the coefficiet of variatio for logormal ad delta-logormal distributios. Curto ad Pito [27] costructed the cofidece iterval for κ whe radom variables are ot idepedetly ad idetically distributed. Recet work of Gulhar et al. [28] has compared several cofidece itervals for estimatig the populatio coefficiet of variatio based o parametric, oparametric, ad modified methods. However,thepopulatiomeamaybekowiseveral pheomea. The cofidece itervals of the aforemetioed authors have ot bee used for estimatig the populatio coefficiet of variatio for the ormal distributio with a kow populatio mea. Therefore, our mai aim i this paper is to propose three cofidece itervals for κ i a ormal distributio with a kow populatio mea. The orgaizatio of this paper is as follows. I Sectio 2, the theoretical backgroud of the proposed cofidece itervals is discussed. The ivestigatios of the performace of the proposed cofidece iterval through a Mote Carlo simulatio study are preseted i Sectio 3. Acompariso of the cofidece itervals is also illustrated by usig a empirical applicatio i Sectio 4.Coclusiosareprovided i the fial sectio. 2. Theoretical Results I this sectio, the mea ad variace of the estimator of the coefficiet of variatio i a ormal distributio with a kow populatio mea are cosidered. I additio, we will itroduce a ubiased estimator for the coefficiet of variatio, obtai its variace, ad fially costruct three cofidece itervals: ormal approximatio, shortest-legth, ad equal-tailed cofidece itervals. If the populatio mea is kow to be μ 0, the the populatio coefficiet of variatio is give by =σ/μ 0.The sample estimate of is = S 0 μ 0, (2) where S 2 0 = 1 i=1 (X i μ 0 ) 2. To fid the expectatio of (2), we have to prove the followig lemma. Lemma 1. Let X 1,X 2,...,X be a radom sample from ormal distributio with kow mea μ 0 ad variace σ 2 ad let S 2 0 = 1 i=1 (X i μ 0 ) 2.The E(S 0 )=c +1 σ, Var (S 0 )=(1 c 2 +1 )σ2, where c +1 = 2/(Γ(( + 1)/2)/Γ(/2)). Proof of Lemma 1. By defiitio, S 2 0 = 1 i=1 (X i μ 0 ) 2 = σ2 where Z i =(X i μ 0 )/σ N(0, 1). Thus, i=1 (3) Z 2 i, (4) S 2 0 σ 2 χ2. (5) Let S 2 = ( 1) 1 i=1 (X i X) 2 ad S 2 = 1 +1 i=1 (X i X) 2. From Theorem B of Rice [29, page197], the distributio of ( 1)S 2 /σ 2 is cetral chi-square distributiowith 1degrees of freedom. Similarly, the distributio of S 2 /σ2 is cetral chi-square distributio with degrees of freedom; that is, S 2 σ 2 χ 2. (6) Oe ca see that [30,page181] E (S) =c σ, (7) where c = 2/( 1)(Γ(/2)/Γ(( 1)/2)). Similarly, E (S ) =c +1 σ, (8) where c +1 = 2/(Γ(( + 1)/2)/Γ(/2)). Equatios (5) ad(6) areequivalet.thus,weobtai E(S 0 ) = E(S ) = c +1 σ.next,wewillfidthevariaceof S 0 : var (S 0 )=E(S 2 0 ) [E(S 0)] 2 =σ 2 c 2 +1 σ2 =(1 c 2 +1 )σ2. (9) By usig Lemma 1, we ca show that the mea ad variace of are E ( ) = c +1 σ =c μ +1, (10) 0 var ( )=( 1 c2 +1 μ0 2 )σ 2 =(1 c 2 +1 )κ2 0. (11)

3 Probability ad Statistics 3 =5 = Coverage probabilities 0.90 Coverage probabilities Vagel Vagel (a) (b) 0.98 = =50 Coverage probabilities Coverage probabilities Vagel Vagel (c) (d) Figure 1: The estimated coverage probabilities of 90% cofidece itervals for the coefficiet of variatio i a ormal distributio with a kow populatio mea. Note that c +1 1as. Therefore, it follows that lim E( )=. (12) It meas that is asymptotically ubiased ad asymptotically cosistet for.from(10),theubiasedestimatorof is = c +1. (13) Usig Lemma 1, the mea ad variace of are give by E ( ) =E( c +1 ) =, (14) var ( )=var ( )= 1 c +1 c+1 2 var ( S 0 ) μ 0 = 1 c 2 +1 μ2 0 (15) (1 c 2 +1 )σ2 =( 1 c2 +1 c+1 2 )κ 2 0.

4 4 Probability ad Statistics =5 = Expected legths Expected legths Vagel Vagel (a) =25 (b) = Expected legths Expected legths Vagel Vagel (c) (d) Figure 2: The expected legths of 90% cofidece itervals for the coefficiet of variatio i a ormal distributio with a kow populatio mea. Thus, lim var ( )=0. (16) Hece, is also asymptotically cosistet for.next,we examie the accuracy of from aother poit view. Let us first cosider the followig theorem. Theorem 2. Let X 1,X 2,...,X be a radom sample from a probability desity fuctio f(x), which has ukow parameter θ.if θ is a ubiased estimator of θ,itcabeshow uder very geeral coditios that the variace of θ must satisfy the iequality var ( θ) 1 E ( 2 / θ 2 l f (x)) = 1 I (θ), (17) where I(θ) is the Fisher iformatio. This is kow as the Cramér-Rao iequality. If var( θ) = 1/(I(θ)), theestimator θ is said to be efficiet. Proof of Theorem 2. See [31, pages ].

5 Probability ad Statistics e e 04 Frequecy 10 Desity 2e (a) Weight 0e Weight (b) Sample quatiles Theoretical quatiles (c) (d) Figure 3: (a) Histogram, (b) desity plot, (c) Box-ad-Whisker plot, ad (d) ormal quatile-quatile plot of the weights of 61 oe-moth old ifats. By settig θ= =σ/μ 0 i Theorem 2,itiseasytoshow that var ( ) κ2 0 2, (18) where is ay ubiased estimator of.thismeasthatthe variace for the efficiet estimator of is κ 2 0 /2. From (15), we will show that (1 c 2 +1 )/c2 +1 1/(2 1). The asymptotic expasio of the gamma fuctio ratio is [32] Γ(j+(1/2)) Γ (j) = j(1 1 8j ). (19) 128j2 Now, if j=/2i (19), we have Thus, we obtai c +1 = 2 Γ ((+1)/2) Γ (/2) = 2 [ 2 ( )] = o ( 1 3/2 ). c 2 +1 = o ( 1 2 ), 1 c 2 +1 c (20) (21)

6 6 Probability ad Statistics Table 1: The values of a ad b for the shortest-legth cofidece iterval for. Cofidece levels df a b a b a b Therefore, var( ) κ 2 0 /(2 1).Thismeasthat is asymptotically efficiet (see (18)). I the followig sectio, three cofidece itervals for are proposed Normal Approximatio Cofidece Iterval. Usig the ormal approximate, we have z= var ( ) = /c +1 (1 c 2 +1 )κ2 0 /c2 +1 = c +1 N (0, 1). 1 c+1 2 (22)

7 Probability ad Statistics 7 Table 2: The estimated coverage probabilities ad expected legths of 90% cofidece itervals for the coefficiet of variatio i a ormal distributio with a kow populatio mea. Coverage probabilities Expected legths Vagel Vagel Therefore, the 100(1 α)% cofidece iterval for based o (22)is Legth Cofidece Iterval. Apivotalquatity for σ 2 is c +1 +z 1 α/2 1 c 2 +1 c +1 z 1 α/2 1 c 2 +1, (23) Q= S2 0 σ 2 χ2. (24) Covertig the statemet where z 1 α/2 is the 100(1 α/2) percetile of the stadard ormal distributio. we ca write P(a S2 0 b)=1 α, (25) σ2 P( b )=1 α. (26) a

8 8 Probability ad Statistics Table 3: The estimated coverage probabilities ad expected legths of 95% cofidece itervals for the coefficiet of variatio i a ormal distributio with a kow populatio mea. Coverage probabilities Expected legths Vagel Vagel Thus, the 100(1 α)% cofidece iterval for basedothe pivotal quatity Q is I order to fid the shortest-legth cofidece iterval for, the followig problem has to be solved: b a, (27) where a, b > 0, a<b, ad the legth of cofidece iterval for is defied as goal: mi ( 1 a,b a 1 b ) costrait: a b f Q (q) dq = 1 α, (29) L= ( 1 a 1 ). (28) b where f Q is the probability desity fuctio of cetral chisquare distributio with degrees of freedom. From Casella

9 Probability ad Statistics 9 Table 4: The 95% cofidece itervals for the coefficiet of variatio of the weight of oe-moth old ifats. Methods Cofidece itervals Lower limit Upper limit Legths Vagel Normal approx ad Berger [33, pages ], the 100(1 α)% shortestlegth cofidece iterval for based o the pivotal quatity Q is determied by the value of a ad b satisfyig b a 3/2 f Q (a) =b 3/2 f Q (b), f Q (q) dq = 1 α. (30) a Table 1 is costructed for the umerical solutios of these equatios by usig the R statistical software [34 36] Equal-Tailed Cofidece Iterval. The 100(1 α)%equaltailed cofidece iterval for basedothepivotalquatity Q is χ 2,1 α/2, χ,α/2 2 (31) where χ 2 ad χ2 are the 100(α/2) ad 100(1 α/2),α/2,1 α/2 percetiles of the cetral chi-square distributio with degrees of freedom, respectively. 3. Simulatio Study A Mote Carlo simulatio was coducted usig the R statistical software [34 36] versio to ivestigate the estimated coverage probabilities ad expected legths of three proposed cofidece itervals ad to compare them to the existig cofidece itervals. The estimated coverage probability ad the expected legth (based o M replicates) are give by 1 α= # (L κ U), M Legth = M j=1 (U j L j ), M (32) where #(L κ U) deotes the umber of simulatio rus for which the populatio coefficiet of variatio κ lies withi the cofidece iterval. The data were geerated from a ormal distributio with a kow populatio mea μ 0 =10 ad = 0.05, 0.10, 0.20, 0.33, 0.50, ad 0.67 ad sample sizes () of 5, 10, 15, 25, 50, ad 100. The umber of simulatio rus (M) isequalto50,000adtheomialcofidecelevels 1 αare fixed at 0.90 ad Three existig cofidece itervals are cosidered, amely, s [7], s [12], ad Vagel s [18]. : : ( z 1 α/2 κ2 0 1 (1 2 + κ2 0 ), ( [( χ2,1 α/2 Vagel: +z 1 α/2 κ2 0 1 (1 2 + κ2 0 )), [( χ2,α/2 ( [( χ2,1 α/2 +2 [( χ2,α/2 +2 1) κ χ2 1/2,1 α/2 1 ], 1) κ χ2 1/2,α/2 1 ] ), 1) κ χ2 1/2,1 α/2 1 ], 1) κ χ2 1/2,α/2 1 ] ). (33) (34) (35) The upper s limit will have to be set to uder the followig coditio [25]: χ 2,α/2 (36) ( 1)( χ,α/2 2 ), ad the upper Vagel s limit will have to be set to uder the followig coditio: χ 2,α/2 (37) ( 1)( χ,α/2 2 2). AscabeseefromTables2 ad 3, the three proposed cofidece itervals have estimated coverage probabilities close to the omial cofidece level i all cases. O the other had, the s, s, ad Vagel s cofidece itervals provide estimated coverage probabilities much differet from the omial cofidece level, especially whe the populatio coefficiet of variatio is large. I other words, the estimated coverage probabilities of existig cofidece itervals ted to be too high. Additioally, the estimated coverage probabilities of existig cofidece itervals icrease as the values of get larger (i.e., for 95% s cofidece iterval, = 10,0.9522for =0.05;0.9539for = 0.10; for =0.67).However,Figure 1 shows that the estimated coverage probabilities of the three proposed cofidece itervals do ot icrease or decrease accordig to the values of. As ca be see from Figure 2, s ad Vagel s cofidece itervals have loger expected legths tha s

10 10 Probability ad Statistics Shapiro-Wilk ormality test data: weight W = 0.978, P-value = Algorithm 1: Shapiro-Wilk test for ormality of the weights of 61 oe-moth old ifats. ad the proposed cofidece itervals. While the expected legths of the three proposed cofidece itervals are shorter tha the legths of the existig oes i almost all cases. Additioally, whe the sample sizes icrease, the legths become shorter (i.e., for 95% shortest-legth cofidece iterval, =0.20,0.1553for=10;0.0949for =25; for =50). 4. A Empirical Applicatio To illustrate the applicatio of the cofidece itervals proposed i the previous sectio, we used the weights (i grams) of 61 oe-moth old ifats listed as follows: (38) The data are take from the study by Ziegler et al. [37](cited i Ledolter ad Hogg [38], page 287). The histogram, desity plot, Box-ad-Whisker plot, ad ormal quatile-quatile plot are displayed i Figure 3. Algorithm 1 shows the result of the Shapiro-Wilk ormality test. As they appear i Figure 3 ad Algorithm 1, wefidthat the data are i excellet agreemet with a ormal distributio. From past research, we assume that the populatio mea of the weight of oe-moth old ifats is about 4400 grams. A ubiased estimator of the coefficiet of variatio is The 95% of proposed ad existig cofidece itervals for the coefficiet of variatio are calculated ad reported i Table 4. This result cofirms that the three cofidece itervals proposed i this paper are more efficiet tha the existig cofidece itervals i terms of legth of iterval. 5. Coclusios The coefficiet of variatio is the ratio of stadard deviatio to the mea ad provides a widely used uit-free measure of dispersio. It ca be useful for comparig the variability betwee groups of observatios. Three cofidece itervals for the coefficiet of variatio i a ormal distributio with a kowpopulatiomeahavebeedeveloped.theproposed cofidece itervals are compared with s, s, ad Vagel s cofidece itervals through a Mote Carlo simulatio study. Normal approximatio, shortest-legth, ad equal-tailed cofidece itervals are better tha the existig cofidece itervals i terms of the expected legth ad the closeess of the estimated coverage probability to the omial cofidece level. Coflict of Iterests The author declares that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmets The author is grateful to Professor Dr. Toghui Wag, Professor Dr. Joh J. Borkowski, ad aoymous referees for their valuable commets ad suggestios, which have sigificatly ehaced the quality ad presetatio of this paper. Refereces [1] K. Kelley, Sample size plaig for the coefficiet of variatio from the accuracy i parameter estimatio approach, Behavior Research Methods,vol.39,o.4,pp ,2007. [2] K. Ah, O the use of coefficiet of variatio for ucertaity aalysis i fault tree aalysis, Reliability Egieerig ad System Safety,vol.47,o.3,pp ,1995. [3] J. Gog ad Y. Li, Relatioship betwee the Estimated Weibull Modulus ad the Coefficiet of Variatio of the Measured Stregth for Ceramics, the America Ceramic Society, vol.82,o.2,pp ,1999. [4]D.S.FaberadH.Kor, Applicabilityofthecoefficietof variatio method for aalyzig syaptic plasticity, Biophysical Joural,vol.60,o.5,pp ,1991. [5]A.J.Hammer,J.R.Stracha,M.M.Black,C.Ibbotso,ad R. A. Elso, A ew method of comparative boe stregth measuremet, Medical Egieerig ad Techology, vol.19,o.1,pp.1 5,1995. [6] J.Billigs,L.Zeitel,J.Lukomik,T.S.Carey,A.E.Blak,ad L. Newma, Impact of socioecoomic status o hospital use i New York City, Health Affairs, vol. 12, o. 1, pp , [7]E.G.adM.J.Karso, Testigtheequalityoftwo coefficiets of variatio, i America Statistical Associatio: Proceedigs of the Busiess ad Ecoomics Sectio, Part I, pp , [8] D.B.Pye,C.B.Trewi,adW.G.Hopkis, Progressioad variability of competitive performace of Olympic swimmers, Sports Scieces,vol.22,o.7,pp ,2004. [9] M. Smithso, Correct cofidece itervals for various regressio effect sizes ad parameters: the importace of ocetral

11 Probability ad Statistics 11 distributios i computig itervals, Educatioal ad Psychological Measuremet,vol.61,o.4,pp ,2001. [10] B. Thompso, What future quatitative social sciece research could look like: cofidece itervals for effect sizes, Educatioal Researcher,vol.31,o.3,pp.25 32,2002. [11] J. H. Steiger, Beyod the F test: effect size cofidece itervals ad tests of close fit i the aalysis of variace ad cotrast aalysis, Psychological Methods, vol. 9, o. 2, pp , [12] A. T., Distributio of the coefficiet of variatio ad the exteded t distributio, the Royal Statistics Society, vol. 95, o. 4, pp , [13] E. C. Fieller, A umerical test of the adequacy of A.T. s approximatio, JouraloftheRoyalStatisticalSociety, vol. 95, o. 4, pp , [14] B. Iglewicz, Some properties of the coefficiet of variatio [Ph.D. thesis], Virgiia Polytechic Istitute, Blacksburg, Va, USA, [15] B. Iglewicz ad R. H. Myers, Comparisos of approximatios to the percetage poits of the sample coefficiet of variatio, Techometrics,vol.12,o.1,pp ,1970. [16] E. S. Pearso, Compariso of A.T. s approximatio with experimetal samplig results, the Royal Statistics Society,vol.95,o.4,pp ,1932. [17] G. J. Umphrey, A commet o s approximatio for the coefficiet of variatio, Commuicatios i Statistics- Simulatio ad Computatio,vol.12,o.5,pp ,1983. [18] M. G. Vagel, Cofidece itervals for a ormal coefficiet of variatio, America Statisticia,vol.50,o.1,pp.21 26, [19] W. Paichkitkosolkul, Improved cofidece itervals for a coefficiet of variatio of a ormal distributio, Thailad Statisticia,vol.7,o.2,pp ,2009. [20] K. K. Sharma ad H. Krisha, Asymptotic samplig distributio of iverse coefficiet-of-variatio ad its applicatios, IEEE Trasactios o Reliability, vol.43,o.4,pp , [21] E. G., Asymptotic test statistics for coefficiet of variatio, Commuicatios i Statistics-Theory ad Methods,vol.20, o. 10, pp , [22] K. C. Ng, Performace of three methods of iterval estimatio of the coefficiet of variatio, IterStat, 2006, pdf. [23] R. Mahmoudvad ad H. Hassai, Two ew cofidece itervals for the coefficiet of variatio i a ormal distributio, Applied Statistics,vol.36,o.4,pp ,2009. [24] L. H. Koopmas, D. B. Owe, ad J. I. Roseblatt, Cofidece itervals for the coefficiet of variatio for the ormal ad logormal distributios, Biometrika,vol.51,o.1-2,pp.25 32, [25] S. Verrill, Cofidece bouds for ormal ad log-ormal distributio coefficiet of variatio, Research Paper EPL-RP- 609, U. S. Departmet of Agriculture, Madiso, Wis, USA, [26] N. Butao ad S. Niwitpog, Cofidece itervals for the differece of coefficiets of variatio for logormal distributios ad delta-logormal distributios, Applied Mathematical Scieces,vol.6,o.134,pp ,2012. [27] J. D. Curto ad J. C. Pito, The coefficiet of variatio asymptotic distributio i the case of o-iid radom variables, Applied Statistics,vol.36,o.1,pp.21 32,2009. [28] M.Gulhar,B.M.G.Kibria,A.N.Albatieh,adN.U.Ahmed, A compariso of some cofidece itervals for estimatig the populatio coefficiet of variatio: a simulatio study, SORT, vol.36,o.1,pp.45 68,2012. [29] J. A. Rice, Mathematical Statistics ad Data Aalysis, Duxbury Press, Belmot, Calif, USA, [30] S. F. Arold, Mathematical Statistics,Pretice-Hall, New Jersey, NJ, USA, [31] E. J. Dudewicz ad S. N. Mishra, Moder Mathematical Statistics, Joh Wiley & Sos, Sigapore, [32] R.L.Graham,D.E.Kuth,adO.Patashik,Aswer to Problem 9.60 i Cocrete Mathematics: A Foudatio for Computer Sciece, Addiso-Wesley, Readig, Pa, USA, [33] G. Casella ad R. L. Berger, Statistical Iferece,DuxburyPress, Califoria,Calif,USA,2001. [34] R. Ihaka ad R. Getlema, R: a laguage for data aalysis ad graphics, JouralofComputatioaladGraphicalStatistics,vol. 5, o. 3, pp , [35] R Developmet Core Team, A Itroductio to R, R Foudatio for Statistical Computig, Viea, Austria, [36] R Developmet Core Team, R: A Laguage ad Eviromet for Statistical Computig, R Foudatio for Statistical Computig, Viea, Austria, [37] E. Ziegler, S. E. Nelso, ad J. M. Jeter, Early Iro Supplemetatio of Breastfed Ifats, Departmet of Pediatrics, Uiversity ofiowa,iowacity,iowa,usa,2007. [38] J. Ledolter ad R. V. Hogg, Applied Statistics for Egieers ad Physical Scietists, Pearso, New Jersey, NJ, USA, 2010.

12 Advaces i Operatios Research Advaces i Decisio Scieces Applied Mathematics Algebra Probability ad Statistics The Scietific World Joural Iteratioal Differetial Equatios Submit your mauscripts at Iteratioal Advaces i Combiatorics Mathematical Physics Complex Aalysis Iteratioal Mathematics ad Mathematical Scieces Mathematical Problems i Egieerig Mathematics Discrete Mathematics Discrete Dyamics i Nature ad Society Fuctio Spaces Abstract ad Applied Aalysis Iteratioal Stochastic Aalysis Optimizatio

Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation

Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece