One-Factor ANOVA Model Using Trapezoidal Fuzzy Numbers Through Alpha Cut Interval Method

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1 nnals of Pure and ppled athematcs Vol., No., 06, 45-6 ISSN: X (P), (onlne) Publshed on January 06 nnals of One-Factor NOV odel sng Trapezodal Fuzzy Numbers Through lpha Cut Interval ethod S. Parthban and P. Gajvaradhan Research Scholar, Department of athematcs, Pachayappa s College, Chenna , Taml Nadu, Inda. Emal: selvam.parthban979@gmal.com Department of athematcs, Pachayappa s College, Chenna , Taml Nadu, Inda. Emal: drgajvaradhan@gmal.com Receved 4 December 05; accepted 4 December 05 bstract. ost of our tradtonal tools n descrptve and nferental statstcs s based on crspness (precseness) of data, measu rements, random varable, hypotheses, and so on. By crsp we mean dchotomous that s, yes-or-no type rather than more-or-less type. But there are many stuatons n whch the above assumptons are rather non-realstc such that we need some new tools to characterze and analyze the problem. By ntroducng fuzzy set theory, dfferent branches of mathematcs are recently studed. But probablty and statstcs attracted more attenton n ths regard because of ther random nature. athematcal statstcs does not have methods to analyze the problems n whch random varables are vague (fuzzy). In ths regard, a smple and new technque for testng the hypotheses under the fuzzy envronments s proposed. Here, the employed data are n terms of trapezodal fuzzy numbers (TFN) whch have been transformed nto nterval data usng α-cut nterval method and on the grounds of the transformed fuzzy data, the one-factor NOV test s executed and decsons are concluded. Ths concept has been llustrated by gvng two numercal examples. Keywords: Fuzzy set, α-cut, Trapezodal fuzzy number (TFN), Test of hypotheses, Onefactor NOV model, pper level data, ower level data. S athematcs Subject Classfcaton (00): 686, 6F03, 97K80. Introducton Fuzzy set theory [35] has been appled to many areas whch need to manage uncertan and vague data. Such areas nclude approxmate reasonng, decson makng, optmzaton, control and so on. In tradtonal statstcal testng [7], the observatons of sample are crsp and a statstcal test leads to the bnary decson. However, n the real lfe, the data sometmes cannot be recorded or collected precsely. The statstcal hypotheses testng under fuzzy envronments has been studed by many authors usng the fuzzy set theory concepts ntroduced by Zadeh [35]. 45

2 S. Parthban and P. Gajvaradhan The applcaton by usng fuzzy set theory to statstcs has been wdely studed n anton et al. [], Buckley [8] and Vertl [9]. rnold [6] proposed the fuzzfcaton of usual statstcal hypotheses and consdered the testng hypotheses under fuzzy constrants on the type I and type II errors. Saade [7], Saade and Schwarzlander [6] consdered the bnary hypotheses testng and dscussed the fuzzy lkelhood functons n the decson makng process by applyng a fuzzfed verson of the Baye s crteron. Grzegorzewsk [4], Watanabe and Imazum [3] proposed the fuzzy test for testng hypotheses wth vague data and the fuzzy test produced the acceptablty of the null and alternatve hypotheses. The statstcal hypotheses testng for fuzzy data by proposng the notons of degrees of optmsm and pessmsm was proposed by Wu [34]. Vertl [9] nvestgated some methods to construct confdence ntervals and statstcal tests for fuzzy data. Wu [33] proposed some approaches to construct fuzzy confdence ntervals for the unknown fuzzy parameter. ref and Taher [5] developed an approach to test fuzzy hypotheses upon fuzzy test statstc for vague data. new approach to the problem of testng statstcal hypotheses s ntroduced by Chach et al. [9]. khko Konsh et al. [5] proposed a method of NOV for the fuzzy nterval data by usng the concept of fuzzy sets. Hypothess testng of one factor NOV model for fuzzy data was proposed by Wu [3] usng the h-level set and the notons of pessmstc degree and optmstc degree by solvng optmzaton problems. Dubos and Prade [] defned any of the fuzzy numbers as a fuzzy subset of the real lne. Chen and Chen [] presented a method for rankng generalzed trapezodal fuzzy numbers. The symmetrc trangular approxmaton was presented by a et al. [0]. Chanas [0] derved a formula for determnng the nterval approxmatons under the Hammng dstance. The trapezodal approxmaton was proposed by bbasbandy et al. [-3]. Grzegorzewsk et al. [5] proposed the trapezodal approxmaton of a fuzzy number, whch s consdered as a reasonable compromse between two opposte tendences: to lose too much nformaton and to ntroduce too sophstcated form of approxmaton from the pont of vew of computaton. In ths paper, we propose a new statstcal fuzzy hypothess testng of NOV model for fndng the sgnfcance among more than two populaton means when the data of ther samples are n terms of trapezodal fuzzy data. We provde the decson rules whch are used to accept or reject the fuzzy null and alternatve hypotheses. In the proposed technque, we convert the gven fuzzy hypothess testng of one factor NOV model wth fuzzy data nto two hypothess testng of one factor NOV models wth crsp data namely, upper level model and lower level model then, we test the hypothess of each of the one factor NOV models wth crsp data and obtan the results and then we obtan a decson about the populaton means on the bass of the proposed decson rules usng the results obtaned. In the decson rules of the proposed testng technque, we are not usng degrees of optmsm, pessmsm and h-level set whch are used n Wu [3]. In fact we would lke to counter an argument that α-cut nterval method s general enough to deal wth one-factor NOV method under fuzzy envronments whch fts better when compared to the smlar problems nvolved under non-fuzzy data. For better understandng, the proposed fuzzy hypothess testng technque of NOV model for fuzzy data s llustrated wth numercal examples. 46

3 One-factor NOV model usng TFNs through alpha cut nterval method. Prelmnares Defnton.. (Generalzed fuzzy number) generalzed fuzzy number s descrbed as any fuzzy subset of the real lne, whose membershp functon μ x satsfes the followng condtons: μ x s a contnuous mappng from to the closed nterval. 0, ω, 0 ω,. μ x = 0, for all x -, a,. μ x x s strctly ncreasng on a, b, v. v. μr x R x s strctly decreasng on v. μ x 0, for all x d,. μ x ω, for all b, c, as ω s a constant and 0 < ω, 47 c, d, where a, b, c, d are real numbers such that a < b c < d. Throughout ths paper, stands for the set of all real numbers, F of fuzzy numbers, expresses a fuzzy number and x x. represents the set ts membershp functon Defnton.. fuzzy set s called normal fuzzy set f there exsts an element (member) x such that μ x. fuzzy set s called convex fuzzy set f where x, x X and α 0, μ αx + - α x mn μ x, μ x set α x X μ x α s sad to be the α - cut of a fuzzy set.. The Defnton.3. fuzzy subset of the real lne wth membershp functonμ such that μ x : 0, convex, x, s called a fuzzy number f s normal, s fuzzy μ x s upper sem-contnuous and Supp s bounded, where Supp cl x : μ x 0 and cl s the closure operator. It s known that for fuzzy number, there exsts four numbers a, b, c, d and two functons x, R x : 0,, where x and R x are nondecreasng and non-ncreasng functons respectvely. Now, we can descrbe a membershp functon as follows: μ x x for a x b; for b x c; R x for c x d; 0 otherwse. x and The functons number respectvely ([, 3]). R x are also called the left and rght sde of the fuzzy

4 In ths paper, we assume that S. Parthban and P. Gajvaradhan x dx < + and t s known that the α - cut of a fuzzy number s α x μ x α, for α 0, and 0 = cl α, α 0, accordng to the defnton of a fuzzy number, t s seen at once that every α - cut of a fuzzy number s a closed nterval. Hence, for a fuzzy number, we have α α, α where α nf x : μ x α and α sup x : μ x α. The left and rght sdes of the fuzzy number are strctly monotone, obvously, and R x respectvely. are nverse functons of x and nother mportant type of fuzzy number was ntroduced n [7] as follows: et a, b, c, d such that a < b c < d. fuzzy number defned as μ x : 0,, n x - a d - x μ x for a x b; for b x c; for c x d; 0 otherwse. b - a d - c a, b, c, d. where n > 0, s denoted by n n x - a nd x b - a, d - x R x d - c the TFN [Dubos and Prade n 98]. n n can also be termed as left and rght spread of a, b, c, d If n, then n n α α, α a + b - a α, d - d - c α ; α 0,. When n = and b = c, we get a trangular fuzzy number. The condtons r =, a = b and c = d mply the closed nterval and n the case r =, a = b = c = d = t (some constant), we can get a crsp number t. Snce a trapezodal fuzzy number s completely characterzed by n = and four real numbers a b c d a, b, c, d. nd the famly of trapezodal, t s often denoted as T fuzzy numbers wll be denoted by F. Now, for n = we have a normal trapezodal fuzzy number a, b, c, d the correspondng α - cut s defned by α a + α b - a, d - α d - c ; α 0,. Now, we need the followng results whch can be found n [7, 9]. and 48

5 One-factor NOV model usng TFNs through alpha cut nterval method Result.. et ntervals on the real lne. D = a, b, a b and a, b, the set of all closed, bounded Result.. et = a, b and B = c, d be n D. Then = B f a = c and b = d. s Result.3. If s the varance of a sample of sze n drawn from the populaton wth ns ns varance σ, then E σ, that s n - n - s an unbased estmator of σ. 3. One-Factor NOV odel The nalyss of Varance (NOV) s a powerful statstcal tool for tests of sgnfcance. The term nalyss of Varance was ntroduced by Prof. R.. Fsher n 90 s to deal wth problems n the analyss of agronomcal data. Varaton s nherent n nature. The total varaton n any set of numercal data s due to a number of causes whch may be classfed as () ssgnable causes and () Chance causes. The varaton due to assgnable causes can be detected and measured whereas the varaton due to chance s beyond the control of human hand and cannot be traced separately. In general, NOV studes manly the homogenety of populatons by separatng the total varance nto ts varous components. That s, ths technque s to test the dfference among the means of populatons by studyng the amount of varaton wthn each of the samples relatve to the amount of varaton between the samples. Samples under employng n NOV model are assumed to be drawn from normal populatons of equal varances. The varaton of each value around ts own grand mean should be ndependent for each value. one-factor NOV s used when the analyss nvolves only one factor wth more than two levels and dfferent subjects n each of the expermental condtons. et a sample of N values of a gven random varable X drawn from a normal populaton wth varance σ whch s subdvded nto h classes accordng to some factor of classfcaton wth whch the classes are homogeneous, that s, there s no dfference between varous classes. th Now, let μ be the mean of populaton class. The test of hypotheses are: Null hypothess: H 0 : μ μ =μh aganst lternatve hypothess: H : μ μ μh. et x j be the value of the th j member of the members. et the general mean of all the N values be x and the mean of th class be x. Now, th class, whch contans n n values n the 49

6 S. Parthban and P. Gajvaradhan x x x x x x x x n x x j j j j j j + n x x where s the sum of the squared devatons of class means from the x x general mean (varaton between classes) and j j s the sum of the squared devatons of varates from the correspondng class means (varaton wthn classes). s total varaton. ns Now, t s known from the theory of estmaton that s an unbased estmate of n- σ, where varance n s s the varance of a sample of sze n drawn from a populaton wth σ. That s, E ns / n- σ. Snce the tems n the th class wth varance n xj x may be consdered as a sample of sze j = n n σ. That s, E xj x σ n n j = n drawn from a populaton wth varance. h.e. E xj x n σ.e. E N - h.e. σ E σ. j = N - h Hence, N - h s an unbased estmate of σ wth N - h degrees of freedom. et us consder the entre group of N tems wth varance x j x as N j the sample of sze N drawn from the same populaton. Now, N E xj x σ.that s, E σ N N j N, ths states that N s an unbased estmate of N degrees of freedom. Now, σ wth E E E N σ N h σ Thus, h - s also an unbased estmate of E σ. h - σ wth h - degrees of freedom. If we assume that the sample drawn from a normal populaton, then the estmates h - and 50

7 One-factor NOV model usng TFNs through alpha cut nterval method N - h are ndependent and hence the rato h - h -, N h N h follows F-dstrbuton wth degrees of freedom. Choosng the rato whch s greater than one, we employ the F-test. For smplcty, let us choose, and. h - N - h ggregatng the above results, the NOV table for one factor classfcaton s gven below([6, 8]): Source of Varaton (S.V.) Sum of Squares (S.S.) Degrees of freedom (d.f.) Between h - Classes Wthn Classes N - h ean Square (.S.) h - N - h Varance Rato (F-value) F = Total N - -- The decson rules of F-test are gven below: () If < and F = Ft where Ft h -, N h s the tabulated value of F wth degrees of freedom at k level of sgnfcance, then we accept the null hypothess H 0, otherwse the alternatve hypothess () If < and F = Ft where Ft N h, h - H s accepted. s the tabulated value of F wth degrees of freedom at k level of sgnfcance, then we accept the null hypothess H 0, otherwse the alternatve hypothess H s accepted. Note that here we use the notaton for level of sgnfcance s to be k nstead of α so as to avod confuson wth α - cut value that can be seen n trapezodal fuzzy numbers (TFN). For smplcty of calculatons, the followng formulae for, and are used: = T xj where T = xj j N j and = -. ; = T T n N where T = x j j 5

8 S. Parthban and P. Gajvaradhan 4. One-Factor NOV model wth TFNs usng α - cut method The fuzzy test of hypotheses of one-factor NOV model where the sample data are trapezodal fuzzy numbers s proposed here. sng the relaton, we transform the fuzzy NOV model to nterval NOV model. Fetchng the upper lmt of the fuzzy nterval, we construct upper level crsp NOV model and consderng the lower lmt of the fuzzy nterval, we construct the lower level crsp NOV model. Thus, n ths proposed approach, two crsp NOV models are desgnated n terms of upper and lower levels. Fnally, we analyse lower level and upper level model usng crsp onefactor NOV technque. et there be N values of samples for a gven random varables X whch are subdvded nto h classes accordng to some knd of classfcaton. Then the lower level data and upper level data for gven trapezodal fuzzy numbers usng α - cut method can be assgned as follows: ower level data: pper level data: a + αb - a a + αb - a a + α b - a a j+ α b j - aj a + α b - a a j+ α b j - a j The one-factor NOV formulae usng α - cut can be tabulated as follows: a + αb - a a + α b - a a j+ α b j - a j where 0 h, 0 j n d- αd - c d - αd - c d - α d - c dj- αd j - cj d - α d - c dj- αd j - cj d - αd - c ower level model T = a j + αb j - aj j N where 0 h, 0 j n. T a j + αb j - aj ; =,,, h. j h nd T = T, T T r n N r = = - d- α d - c dj- αd j - cj where 0 h, 0 j n 5 pper level model T d j - αd j - cj j N where 0 h, 0 j n. = T j j j j d - α d - c ; =,,..., h h nd T = T, T T r n N r = = -

9 One-factor NOV model usng TFNs through alpha cut nterval method et k be the level of sgnfcance. Now, the null hypothess: H 0 : μ μ μh aganst the alternatve hypothess: H : μ μ μ h H 0 : μ μ μ h aganst H : μ μ μ h. H 0, H 0 : μ, μ μ, μ = μ h, μ h aganst H, H : μ, μ μ, μ μ h, μ h The followng two sets of hypotheses can be obtaned. () () The null hypothess hypothess H : μ μ μ. The null hypothess hypothess h H : μ μ μ. H : μ μ μ aganst the alternatve 0 h H : μ μ μ aganst the alternatve h 0 h Decson rules: () If F F t () at k level of sgnfcance wth N - h, h - freedom then the null hypothess H0 α 0,, otherwse the alternatve hypothess If F F t degrees of s accepted for certan value of H s accepted. at k level of sgnfcance wth N - h, h - freedom then the null hypothess H0 α 0,, otherwse the alternatve hypothess degrees of s accepted for certan value of H s accepted. Example 4.. food company wshed to test four dfferent package desgns for a new product. Ten stores wth approxmately equal sales volumes are selected as the expermental unts. Package desgns and 4 are assgned to three stores each and package desgns and 3 are assgned to two stores each. We cannot record the exact sales volume n a store due to some unexpected stuatons, but we have the fuzzy data for sales volumes. The fuzzy data are gven below [3]: Package desgn () Store (Observaton j) 3 4, 5, 7, , 0,, 3, 3, 6, 9 0, 4, 6, 0,, 4, 5 3 5, 7, 9, 4, 6, 9, 0 7, 0,, 3 4 5, 8,, 3, 3, 5,

10 S. Parthban and P. Gajvaradhan We test the hypothess whether the fuzzy mean sales are same for four desgns of package or not. et μ be the mean sales for the th desgn. Then the null hypothess H : μ μ μ μ aganst the alternatve hypothess H : μ μ μ μ Now, the nterval model for the gven trapezodal fuzzy number usng α - cut method s: Package desgn () Now, the NOV tables for lower level α - cut nterval and upper level α - cut nterval are gven below: ower level model: The null hypothess H : μ μ μ μ. 3 4 pper level model: Store (Observaton j) α, 8 - α α, 3 - α + α, 9-3α 0 + 4α, 0-4α + α, 5 - α α, - α 4 + α, 0 - α 7 + 3α, 3 - α α, 3 - α + α, 7 - α -- Package desgn () H : μ μ μ μ aganst the alternatve hypothess Store (Observaton j) α α + α 0 + 4α + α α 4 + α 7 + 3α α + α -- Package desgn () Store (Observaton j) α α 9-3α 0-4α 5 - α 3 - α 0 - α 3 - α α 7 - α -- 54

11 One-factor NOV model usng TFNs through alpha cut nterval method The null hypothess H : μ μ μ μ. 3 4 H : μ μ μ μ aganst the alternatve hypothess The NOV table for lower level model: Source of Varance (S.V.) Between Classes Wthn Classes Sum of Squares (S.S.) Degrees of freedom (d.f.) h - = 4 3 N - h = ean Square (.S.) 3 6 F-rato F C C F Here, N = 0 and n, 3, 3, for the package desgns,, 3, 4 respectvely. T = 37 + α ; T 83α α n 6 and a j+ α bj- a j 53α + 584α j 89α + 86α ; 84α + 876α and 350α - 360α nd 84α + 876α ; 350α - 360α and 0 α and FC 4 46α + 9α + 34 C F 5 35α - 36α + 5 where s the calculated value of F at lower level model. Now, the tabulated value of F at k = 5% level of sgnfcance wth h -, N - h 3, 6 degrees of freedom s Ft at 5% Here, FC F t at α = 0.and F C > F t for 0. α. Hence, the null hypothess H0 s rejected at 5% level of sgnfcance for 0. α. 55

12 The NOV table for upper level model: Source of Varance (S.V.) Between Classes Wthn Classes Sum of Squares (S.S.) Here, N = 0 and n, 3, 3, S. Parthban and P. Gajvaradhan Degrees of freedom (d.f.) h - = FC N - h = for the package desgns,, 3, 4 respectvely. ean Square (.S.) F-rato F C T = 99-9α ; T 38α α n 6 and j j j j d - α d - c 45α - 78α ; 07α - 4α and 89α - 58α α - α nd 07α - 4α ; 3α - α and 0 α and FC 07α - 4α + 33 C F 5 3α - α + 35 where s the calculated value of F at upper level model. Now, the tabulated value of F at k = 5% level of sgnfcance wth h -, N - h 3, 6 degrees of freedom s Ft at 5% Here, F C > F t for all α where 0 α. Hence we reject the null hypothess H0 at 5% level of sgnfcance for all α 0 α. Thus, the rejecton level of null hypotheses for lower and upper level data are gven below: H0 s rejected for all α; 0. α and H0 s rejected for all α; 0 α. Therefore, we accept the alternatve hypothess H of the fuzzy NOV model. Concluson 4.. The factor level fuzzy means μ are not equal. Hence, we conclude that there s a relaton between package desgn and sales volumes. Remark 4.. In ths proposed method, the notons of pessmstc degree and optmstc degree are not used. The whole calculaton technque s fully based on α - cut nterval 56

13 One-factor NOV model usng TFNs through alpha cut nterval method method [4]. nd the decson obtaned n the proposed fuzzy hypothess testng usng α - cut nterval NOV method for example- fts better when compared wth Wu [3]. Example 4.. In order to determne whether there s sgnfcant dfference n the durablty of 3 makes of computers, samples of sze 5 are selected from each make and the frequency of repar durng the frst year of purchase s observed. The results are obtaned n terms of fuzzy data due to dfferent knds of mantenance and usage. The results are as follows: akes B C 4, 6, 8, 9 3, 5, 7, 8 6, 8, 0, 3 4, 6, 9, 0 8, 9,,, 4, 5, 7 6, 8, 0, 9,, 3, 5, 5, 7, 9 8, 0,, 4 9,, 4, 5, 5, 8, 0 5, 7, 9,, 4, 6, 9,, 4, 7 In vew of the above data, the testng procedure s proposed to check s there any sgnfcant dfference n the durablty of the 3 makes of computers? We test the hypothess whether the fuzzy means of the 3 makes of computers dffer or not. Here, the null hypothess H 0 : μ μ μ3 aganst the alternatve hypothess H : μ μ μ. 3 Now, the NOV model usng α - cut nterval method for gven fuzzy data s tabulated below: Sample (Observaton j) ake , 8-4+, 0-6+, - 8+, 4-5+, -3 B 6+, , - 9+, 5-9+3, 5- +, 9-3 C 4+, 9- +, 7- +3, 9- +3, 0- +, 7-3 The NOV tables for ower level α - cut nterval and pper level α - cut nterval are gven below: 57

14 ower level α - cut nterval: The null hypothess H : μ μ μ. 3 pper level α - cut nterval: S. Parthban and P. Gajvaradhan Sample (Observaton j) ake B C H : μ μ μ aganst the alternatve hypothess 0 3 Sample (Observaton j) ake B C The null hypothess H : μ μ μ. 3 H : μ μ μ aganst the alternatve hypothess 0 3 The NOV table for lower level model: S.V. S.S. d.f..s. F-rato F C Between Classes Wthn Classes h - = 3 N - h = 5 3 C F Here, N = 5 and n 5, 5, 5 T = 7 + 3α ; for the makes, B, C respectvely. T 3α + 44α n 5 and j j j j a + α b - a 69α + 9α ; α - 76α and 74α - α

15 One-factor NOV model usng TFNs through alpha cut nterval method 4α + 8α nd α - 76α ; 4α + 8α and α - 76α + 88 FC α + 9α + 36 where 0 α and F C s the calculated value of F at lower level model. Now, the tabulated value of F at h -, N - h, degrees of freedom s k = 5% level of sgnfcance wth F Snce, F > F α, 0 α t at 5% C t at 5%, we reject the null hypothess H 0. There s a sgnfcant dfference n the durablty of the 3 makes of computers at lower level of α - cut. The NOV table for upper level model: S.V. S.S. d.f..s. F-rato F C Between Classes Wthn Classes h - = 3 N - h = 5 3 C F Here, N = 5 and n 5, 5, 5 T = 6-8α ; for the makes, B, C respectvely. T 64α α n 5 and j j j j d - α d - c 6α - 596α α + 76α ; 8α + 6α and 46α + 0α nd 8α + 6α ; 46α + 0α and 8α + 6α FC 3α + 0α + 30 where 0 α and FC s the calculated value of F at upper level model. nd the tabulated value of F at k = 5% level of sgnfcance wth h -, N - h, F Here, F > F α, 0 α t at 5% C t at 5% degrees of freedom s, we reject the null hypothess H 0. There s a sgnfcant dfference n the durablty of the 3 makes of computers at upper level of α - cut. 59

16 5. Concluson Therefore, the null hypotheses S. Parthban and P. Gajvaradhan H0 and H are rejected α, 0 α 0. We conclude n general that there s a sgnfcant dfference between n the durablty of the 3 makes of computers. REFERENCES. S. bbasbandy and B. sady, The nearest trapezodal fuzzy number to a fuzzy quantty, ppl. ath. Comput., 56 (004) S. bbasbandy and. mrfakhran, The nearest approxmaton of a fuzzy quantty n parametrc form, ppl. ath. Comput., 7 (006) S. bbasbandy and. mrfakhran, The nearest trapezodal form of generalzed left rght fuzzy number, Internat. J. pprox. Reason., 43 (006) S. bbasbandy and T. Hajjar, Weghted trapezodal approxmaton-preservng cores of a fuzzy number, Comput, and aths. wth pplcatons, 59 (00) ref and S.. Taher, Testng fuzzy hypotheses usng fuzzy data based on fuzzy test statstc, Journal of ncertan Systems, 5 (0) B. F. rnold, Testng fuzzy hypotheses wth crsp data, Fuzzy Sets and Systems, 94 (998) S. Bodjanova, edan value and medan nterval of a fuzzy number, Inform. Sc., 7 (005) J. J. Buckley, Fuzzy statstcs, Sprnger-Verlag, New York, J. Chach, S.. Taher and R. Vertl, Testng statstcal hypotheses based on fuzzy confdence ntervals, Forschungsbercht S-0-, Technsche nverstat Wen, ustra, S. Chanas, On the nterval approxmaton of a fuzzy number, Fuzzy Sets and Systems, 0 (999) -6.. S. Chen and G., Representaton, rankng, and dstance of fuzzy number wth exponental membershp functon usng graded mean ntegraton method, Tamsu, Oxf. J. ath. Sc., 6 () (000) D. Dubos and H. Prade, Operatons on fuzzy numbers, Int. J. Syst. Sc., 9 (978) D. Dubos and H. Prade, Fuzzy sets and systems: Theory and applcaton, cademc Press, New York, P. Grzegorzewsk, Testng statstcal hypotheses wth vague data, Fuzzy Sets and Systems, (000) P. Grzgorzewsk and E. rowka, Trapezodal approxmaton of fuzzy numbers, Fuzzy Sets and Systems, 53 (005) S. C. Gupta and V. K. Kapoor, Fundamentals of mathematcal statstcs, Sultan Chand & Sons, New Delh, Inda. 7. R. R. Hockng, ethods and applcatons of lnear models: regresson and the analyss of varance, New York: John Wley & Sons, D. Kalpanaprya and P. Pandan, Fuzzy hypothess testng of NOV model wth fuzzy data, Int. Nat. J. odern Engg. Research, (4) (0)

17 One-factor NOV model usng TFNs through alpha cut nterval method 9. George J. Klr and Bo Yuan, Fuzzy sets and fuzzy logc, Theory and pplcatons, Prentce-Hall, New Jersey, a,. Kandel and. Fredman, new approach for defuzzfcaton, Fuzzy Sets and Systems, (000) Pal and H. Rashmanlou, Irregular nterval valued fuzzy graphs, nnals of Pure and ppled athematcs, 3() (03) K. G. anton,.. Woodbury and H. D. Tolley, Statstcal applcatons usng fuzzy sets, New York: John Wley & Sons, G. Voskoglou, trangular fuzzy model for assessng problem solvng sklls, nnals of Pure and ppled athematcs, 7() (04) G. Voskoglou, n applcaton of fuzzy logc to computatonal thnkng, nnals of Pure and ppled athematcs, () (0) khko Konsh, Tetsuj Okuda and K. sa, nalyss of varance based on fuzzy nterval data usng moment correcton method, Internatonal Journal of Innovatve Computng, (006) J. J. Saade and H. Schwarzlander, Fuzzy hypothess testng wth hybrd data, Fuzzy Sets and Systems, 35 (990) J. J. Saade, Extenson of fuzzy hypothess testng wth hybrd data, Fuzzy Sets and Systems, 63 (994) T. Veerarajan, Probablty, statstcs and random process, Tata cgraw Hll Educaton Pvt. td., New Delh, Inda. 9. R. Vertl, nvarate statstcal analyss wth fuzzy data, Computatonal Statstcs and Data nalyss, 5 (006) R. Vertl, Statstcal methods for fuzzy data, John Wley and Sons, Chchester, N. Watanabe and T. Imazum, fuzzy statstcal test of fuzzy hypotheses, Fuzzy Sets and Systems, 53 (993) H. C. Wu, nalyss of varance for fuzzy data, Internatonal Journal of Systems Scence, 38 (007) H. C. Wu, Statstcal confdence ntervals for fuzzy data, Expert Systems wth pplcatons, 36 (009) H. C. Wu, Statstcal hypotheses testng for fuzzy data, Informaton Scences, 75 (005) Zadeh, Fuzzy sets, Informaton and Control, 8 (965)