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

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1 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg Oe-Factor NOV odel sg Trapezodal Fuzzy Numbers Through lpha ut Iterval ethod bstract P Gajvaradha ad S Parthba * Departmet of athematcs, Pachayappa s ollege, hea-6 3, Taml Nadu, Ida Emal: Research Scholar, Departmet of athematcs, Pachayappa s ollege, hea-6 3, Taml Nadu, Ida ost of our tradtoal tools descrptve ad feretal statstcs s based o crspess (precseess) of data, measuremets, radom varable, hypotheses, ad so o By crsp we mea dchotomous that s, yes-or-o type rather tha more-or-less type But there are may stuatos whch the above assumptos are rather orealstc such that we eed some ew tools to characterze ad aalyze the problem By troducg fuzzy set theory, dfferet braches of mathematcs are recetly studed But probablty ad statstcs attracted more atteto ths regard because of ther radom ature athematcal statstcs does ot have methods to aalyze the problems whch radom varables are vague (fuzzy) I ths regard, a smple ad ew techque for testg the hypotheses uder the fuzzy evromets s proposed ere, the employed data are terms of trapezodal fuzzy umbers (TFN) whch have bee trasformed to terval data usg α-cut terval method ad o the grouds of the trasformed fuzzy data, the oe-factor NOV test s executed ad decsos are cocluded Ths cocept has bee llustrated by gvg two umercal examples Keywords: Fuzzy set, α-cut, Trapezodal fuzzy umber (TFN), Test of hypotheses, Oe-factor NOV model, pper level data, ower level data Itroducto Fuzzy set theory [3] has bee appled to may areas whch eed to maage ucerta ad vague data Such areas clude approxmate reasog, decso makg, optmzato, cotrol ad so o I tradtoal statstcal testg [7], the observatos of sample are crsp ad a statstcal test leads to the bary decso owever, the real lfe, the data sometmes caot be recorded or collected precsely The statstcal hypotheses testg uder fuzzy evromets has bee studed by may authors usg the fuzzy set theory cocepts troduced by Zadeh [3] The applcato by usg fuzzy set theory to statstcs has bee wdely studed ato et al [] ad Buckley [8] ad Vertl [7] rold [6] proposed the fuzzfcato of usual statstcal hypotheses ad cosdered the testg hypotheses uder fuzzy costrats o the type I ad type II errors Saade [4], Saade ad Schwarzlader [3] cosdered the bary hypotheses testg ad dscussed the fuzzy lkelhood fuctos the decso makg process by applyg a fuzzfed verso of the Baye s crtero Grzegorzewsk [4] ad Wataabe ad Imazum [8] proposed the fuzzy test for testg hypotheses wth vague data ad the fuzzy test produced the acceptablty of the ull ad alteratve hypotheses The statstcal hypotheses testg for fuzzy data by proposg the otos of degrees of optmsm ad pessmsm was proposed by Wu [3] Vertl [6] vestgated some methods to costruct cofdece tervals ad statstcal tests for fuzzy data Wu [3] proposed some approaches to costruct fuzzy cofdece tervals for the ukow fuzzy parameter ref ad Taher [5] developed a approach to test fuzzy hypotheses upo fuzzy test statstc for vague data ew approach to the problem of testg statstcal hypotheses s troduced by hach et al [9] khko Kosh et al [] proposed a method of NOV for the fuzzy terval data by usg the cocept of fuzzy sets ypothess testg of oe factor NOV model for fuzzy data was proposed by Wu [9] usg the h-level set ad the otos of pessmstc degree ad optmstc degree by solvg optmzato problems Dubos 87

2 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg ad Prade [] defed ay of the fuzzy umbers as a fuzzy subset of the real le he ad he [] preseted a method for rakg geeralzed trapezodal fuzzy umbers The symmetrc tragular approxmato was preseted by a et al [] haas [] derved a formula for determg the terval approxmatos uder the ammg dstace The trapezodal approxmato was proposed by bbasbady et al [-3] Grzegorzewsk et al [5] proposed the trapezodal approxmato of a fuzzy umber, whch s cosdered as a reasoable compromse betwee two opposte tedeces: to lose too much formato ad to troduce too sophstcated form of approxmato from the pot of vew of computato I ths paper, we propose a ew statstcal fuzzy hypothess testg of NOV model for fdg the sgfcace amog more tha two populato meas whe the data of ther samples are terms of trapezodal fuzzy data We provde the decso rules whch are used to accept or reject the fuzzy ull ad alteratve hypotheses I the proposed techque, we covert the gve fuzzy hypothess testg of oe factor NOV model wth fuzzy data to two hypothess testg of oe factor NOV models wth crsp data amely, upper level model ad lower level model the, we test the hypothess of each of the oe factor NOV models wth crsp data ad obta the results ad the we obta a decso about the populato meas o the bass of the proposed decso rules usg the results obtaed I the decso rules of the proposed testg techque, we are ot usg degrees of optmsm, pessmsm ad h-level set whch are used Wu [9] I fact we would lke to couter a argumet that α-cut terval method s geeral eough to deal wth oe-factor NOV method uder fuzzy evromets whch fts better whe compared to the smlar problems volved uder o-fuzzy data For better uderstadg, the proposed fuzzy hypothess testg techque of NOV model for fuzzy data s llustrated wth umercal examples Prelmares Defto Geeralzed fuzzy umber geeralzed fuzzy umber s descrbed as ay fuzzy subset of the real le μ x satsfes the followg codtos: μ x s a cotuous mappg from to the closed terval μ x =, for all x -, a, μ x x s strctly creasg o a, b, v μ x ω, for all b, c, as ω s a costat ad < ω, v μr x R x s strctly decreasg o c, d, v μ x, for all x d, where a, b, c, d are real umbers such that a < b c < d, whose membershp fucto, ω, ω, Throughout ths paper, stads for the set of all real umbers, F represets the set of fuzzy umbers, expresses a fuzzy umber ad Defto x ts membershp fucto x fuzzy set s called ormal fuzzy set f there exsts a elemet (member) x such that fuzzy set s called covex fuzzy set f μ αx + - α x mμ x, μ x x, x X ad α, The set α x X μ x α Defto 5 μ x where s sad to be the α - cut of a fuzzy set 88

3 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg fuzzy subset of the real le wth membershp fucto μ x such that μ x :, called a fuzzy umber f s ormal, s fuzzy covex, bouded, where μ x s upper sem-cotuous ad Supp cl x : μ x ad cl s the closure operator, s Supp s It s kow that for fuzzy umber, there exsts four umbers a, b, c, d ad two fuctos, where x, R x :, x ad R x are o-decreasg ad o-creasg fuctos respectvely Now, we ca descrbe a membershp fucto as follows: x ad μ x x for a x b; for b x c; R x for c x d; otherwse The fuctos R x are also called the left ad rght sde of the fuzzy umber respectvely ([, 3]) I ths paper, we assume that x dx < + ad t s kow that the α - cut of a fuzzy umber s α x μ x α, for α, ad = cl α, accordg to the defto of a fuzzy α, umber, t s see at oce that every α - cut of a fuzzy umber s a closed terval ece, for a fuzzy umber α α, α α f x : μ x α ad, we have where α sup x : μ x α The left ad rght sdes of the fuzzy umber are strctly mootoe, obvously, fuctos of R x respectvely x ad other mportat type of fuzzy umber was troduced [7] as follows: et a, b, c, d such that a < b c < d ad fuzzy umber defed as μ x :, are verse, x - a d - x μ x for a x b; for b x c; for c x d; otherwse where b - a d - c x - a >, s deoted by a, b, c, d d x b - a termed as left ad rght spread of the TFN [Dubos ad Prade 98] a, b, c, d If, the ad α α, α a + b - a α, d - d - c α ; α, R x d - x d - c ca also be Whe = ad b = c, we get a tragular fuzzy umber The codtos r =, a = b ad c = d mply the closed terval ad the case r =, a = b = c = d = t (some costat), we ca get a crsp umber t Sce a trapezodal fuzzy umber s completely characterzed by = ad four real umbers a b c d, t s T a, b, c, d F ofte deoted as d the famly of trapezodal fuzzy umbers wll be deoted by 89

4 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg Now, for = we have a ormal trapezodal fuzzy umber a, b, c, d s defed by α a + αb - a, d - αd - c ; α, ad the correspodg α - cut Now, we eed the followg results whch ca be foud [7, 9] Result et D = a, b, a b ad a, b, the set of all closed, bouded tervals o the real le Result et = a, b ad B = c, d be D The = B f a = c ad b = d Result 3 If s s the varace of a sample of sze draw from the populato wth varace that s s - s a ubased estmator of σ 3 Oe-Factor NOV odel σ, the s E σ - The alyss of Varace (NOV) s a powerful statstcal tool for tests of sgfcace The term alyss of Varace was troduced by Prof R Fsher 9 s to deal wth problems the aalyss of agroomcal data Varato s heret ature The total varato ay set of umercal data s due to a umber of causes whch may be classfed as () ssgable causes ad () hace causes The varato due to assgable causes ca be detected ad measured whereas the varato due to chace s beyod the cotrol of huma had ad caot be traced separately I geeral, NOV studes maly the homogeety of populatos by separatg the total varace to ts varous compoets That s, ths techque s to test the dfferece amog the meas of populatos by studyg the amout of varato wth each of the samples relatve to the amout of varato betwee the samples Samples uder employg NOV model are assumed to be draw from ormal populatos of equal varaces The varato of each value aroud ts ow grad mea should be depedet for each value oe-factor NOV s used whe the aalyss volves oly oe factor wth more tha two levels ad dfferet subjects each of the expermetal codtos et a sample of N values of a gve radom varable X draw from a ormal populato wth varace σ whch s subdvded to h classes accordg to some factor of classfcato wth whch the classes are homogeeous, that s, there s o dfferece betwee varous classes th Now, let μ be the mea of populato class The test of hypotheses are: Null hypothess: :μ μ =μh agast lteratve hypothess: :μ μ μh et xj be the value of the the N values be x ad the mea of th j member of the values the th class, whch cotas th class be x Now, members et the geeral mea of all j j j j j j + x x x x x x x x x x, 9

5 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg x x where (varato betwee classes) ad s the sum of the squared devatos of class meas from the geeral mea x x j j the correspodg class meas (varato wth classes) s total varato Now, t s kow from the theory of estmato that s - varace of a sample of sze draw from a populato wth varace the tems the from a populato wth varace s the sum of the squared devatos of varates from s a ubased estmate of th class wth varace xj x j = σ That s, E xj x σ j = σ, where σ That s, Es / - σ s s the Sce may be cosdered as a sample of sze draw h e E xj x σ e E N - h e σ E σ j = N - h ece, N - h s a ubased estmate of σ wth N - h degrees of freedom et us cosder the etre xj x as the sample of sze N draw from the same populato N group of N tems wth varace N E xj x σ N N j Now, ubased estmate of j That s, σ wth N E E E N σ N h σ Thus, E σ N, ths states that N s a degrees of freedom Now, E σ h - h - s also a ubased estmate of σ wth h - degrees of freedom If we assume that the sample draw from a ormal populato, the the estmates depedet ad hece the rato h - N h follows F-dstrbuto wth h - ad N - h are h -, N h degrees of freedom hoosg the rato whch s greater tha oe, we employ the F-test For smplcty, let us choose, ad N - h ggregatg the above results, the NOV table for oe factor classfcato s gve below ([6, 5]): h - 9

6 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg Source of Varato (SV) Sum of Squares (SS) Degrees of freedom (df) Betwee lasses h - Wth lasses N - h ea Square (S) h - N - h Varace Rato (Fvalue) F = Total N - -- The decso rules of F-test are gve below: () If < ad F = F t where t F s the tabulated value of F wth h -, N h degrees of freedom at k level of sgfcace, the we accept the ull hypothess, otherwse the alteratve hypothess () If < ad s accepted F = F t where t F s the tabulated value of F wth N h, h - degrees of freedom at k level of sgfcace, the we accept the ull hypothess, otherwse the alteratve hypothess s accepted Note that here we use the otato for level of sgfcace s to be k stead of α so as to avod cofuso wth α - cut value that ca be see trapezodal fuzzy umbers (TFN) For smplcty of calculatos, the followg formulae for, ad are used: = T xj where T = xj j N j = - ; = T 4 Oe-Factor NOV model wth TFNs usg α - cut method: T N where T = x j ad The fuzzy test of hypotheses of oe-factor NOV model where the sample data are trapezodal fuzzy umbers s proposed here sg the relato, we trasform the fuzzy NOV model to terval NOV model Fetchg the upper lmt of the fuzzy terval, we costruct upper level crsp NOV model ad cosderg the lower lmt of the fuzzy terval, we costruct the lower level crsp NOV model Thus, ths proposed approach, two crsp NOV models are desgated terms of upper ad lower levels Fally, we aalyse lower level ad upper level model usg crsp oe-factor NOV techque et there be N values of samples for a gve radom varables X whch are subdvded to h classes accordg to some kd of classfcato The the lower level data ad upper level data for gve trapezodal fuzzy umbers usg α - cut method ca be assged as follows: j 9

7 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg ower level data: pper level data: a + αb - a a + αb - a a + α b - a a j+ αb j - a j a + α b - a a j+ αb j - a j a + αb - a a + α b - a a j+ αb j - a j where h, j d- αd - c d - αd - c d - α d - c dj- αd j - c j d - α d - c dj- αd j - c j d - αd - c d- α d - c dj- αd j - c j where h, j The oe-factor NOV formulae usg α - cut ca be tabulated as follows: ower level model T = a j + αb j - aj j N where h, j T a j + αb j - aj ; =,,,h j h d T = Tr, r = = - T T N pper level model T d j - αd j - cj j N where h, j T d j - αd j - cj ; =,,,h = j h d T = Tr, r = = - T T N et k be the level of sgfcace Now, the ull hypothess: : μ μ μh agast the alteratve hypothess: : μ μ μh : μ μ μ h : μ μ μ agast h, : μ, μ μ, μ = μ h, μ h agast, : μ, μ μ, μ μ h, μ h The followg two set of hypotheses ca be obtaed () The ull hypothess :μ μ μ h : μ μ μ agast the alteratve hypothess h 93

8 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg () The ull hypothess Decso rules: () () Example 4 :μ μ μ If h F F t hypothess : μ μ μ agast the alteratve hypothess h at k level of sgfcace wth N - h, h - s accepted for certa value of α, s accepted If F F t hypothess at k level of sgfcace wth N - h, h - s accepted for certa value of α, s accepted degrees of freedom the the ull, otherwse the alteratve hypothess degrees of freedom the the ull, otherwse the alteratve hypothess food compay wshed to test four dfferet package desgs for a ew product Te stores wth approxmately equal sales volumes are selected as the expermetal uts Package desgs ad 4 are assged to three stores each ad package desgs ad 3 are assged to two stores each We caot record the exact sales volume a store due to some uexpected stuatos, but we have the fuzzy data for sales volumes The fuzzy data are gve below [9]: Package desg () Store (Observato j) 3 9,,, 3 4, 5, 7, 8 --, 3, 6, 9, 4, 6,,, 4, 5 3 5, 7, 9, 4, 6, 9, 7,,, 3 4 5, 8,, 3, 3, 5, 7 -- We test the hypothess whether the fuzzy mea sales are same for four desgs of package or ot et mea sales for the hypothess : μ μ μ3 μ4 th desg The the ull hypothess 3 4 μ be the : μ μ μ μ agast the alteratve Now, the terval model for the gve trapezodal fuzzy umber usg α - cut method s: Package desg () Store (Observato j) α, 3 - α 4 + α, 8 - α -- + α, 9-3α + 4α, - 4α + α, 5 - α α, - α 4 + α, - α 7 + 3α, 3 - α α, 3 - α + α, 7 - α -- Now, the NOV tables for lower level α - cut terval ad upper level α - cut terval are gve below: 94

9 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg ower level model: Package desg () Store (Observato j) α 4 + α -- + α + 4α + α α 4 + α 7 + 3α α + α -- The ull hypothess pper level model: : μ μ μ μ agast the alteratve hypothess 3 4 :μ μ μ μ 3 4 Package desg () The ull hypothess :μ μ μ μ 3 4 The NOV table for lower level model: Source of Varace (SV) Betwee lasses Wth lasses Store (Observato j) α 8 - α α - 4α 5 - α 3 - α - α 3 - α α 7 - α -- Sum of Squares (SS) : μ μ μ μ agast the alteratve hypothess 3 4 Degrees of freedom (df) h - = 4 3 N - h = 4 6 ea Square (S) ere, N = ad, 3, 3, for the package desgs,, 3, 4 respectvely F-rato F 3 F 6 T = 37 + α ; T 83α + 354α ad 6 a j+ α bj- a j 53α + 584α j 89α + 86α + 8 ; 84α + 876α ad 95

10 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg 35α - 36α d 84α + 876α ; 35α - 36α ad 4 46α + 9α + 34 F 5 35α - 36α + 5 where α ad F s the calculated value of F at lower level model Now, the tabulated value of F at k = 5% level of sgfcace h -, N - h 3, 6 degrees of freedom s F 476 ere, F F at α = ad wth F > F for α t ece, the ull hypothess The NOV table for upper level model: Source of Varace (SV) Betwee lasses Wth lasses t at 5% s rejected at 5% level of sgfcace for α Sum of Squares (SS) Degrees of freedom (df) h - = 4 3 N - h = 4 6 ere, N = ad, 3, 3, for the package desgs,, 3, 4 respectvely ea Square (S) t F-rato F 3 F 6 T = 99-9α ; T 38α - 458α ad 6 j dj- α dj- cj 45α - 78α + 47 ; 7α - 4α ad 89α - 58α α - α d 7α - 4α ; 3α - α ad 7α - 4α + 33 F 5 3α - α + 35 where α ad F s the calculated value of F at upper level model Now, the tabulated value of F at k = 5% level of sgfcace wth h -, N - h 3, 6 degrees of freedom s F 476 ere, F > F for all α where α ece we reject the ull hypothess t at 5% 96 at 5% level of sgfcace for all α α Thus, the rejecto level of ull hypotheses for lower ad upper level data are gve below: s rejected for all α; α ad Therefore, we accept the alteratve hypothess ocluso 4 s rejected for all α; α of the fuzzy NOV model The factor level fuzzy meas μ are ot equal ece, we coclude that there s a relato betwee package desg ad sales volumes t

11 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg Remark 4 I ths proposed method, the otos of pessmstc degree ad optmstc degree are ot used The whole calculato techque s fully based o α - cut terval method [4] d the decso obtaed the proposed fuzzy hypothess testg usg α - cut terval NOV method for example- fts better whe compared wth Wu [9] Example 4 I order to determe whether there s sgfcat dfferece the durablty of 3 makes of computers, samples of sze 5 are selected from each make ad the frequecy of repar durg the frst year of purchase s observed The results are obtaed terms of fuzzy data due to dfferet kds of mateace ad usage The results are as follows: akes B 3, 5, 7, 8 6, 8,, 3 4, 6, 8, 9 4, 6, 9, 8, 9,,, 4, 5, 7 6, 8,, 9,, 3, 5, 5, 7, 9 8,,, 4 9,, 4, 5, 5, 8, 5, 7, 9,, 4, 6, 9,, 4, 7 I vew of the above data, the testg procedure s proposed to check s there ay sgfcat dfferece the durablty of the 3 makes of computers? We test the hypothess whether the fuzzy meas of the 3 makes of computers dffer or ot ere, the ull hypothess : μ μ μ3 agast the alteratve hypothess : μ μ μ3 ake Now, the NOV model usg α - cut terval method for gve fuzzy data s tabulated below: Sample (Observato j) α, 8 - α 4 + α, - α 6 + α, - α 8 + α, 4 - α 5 + α, - 3α B 6 + α, 3-3α 8 + α, - α 9 + α, 5 - α 9 + 3α, 5 - α + α, 9-3α 4 + α, 9 - α + α, 7 - α + 3α, 9 - α + 3α, - α + α, 7-3α The NOV tables for ower level α - cut terval ad pper level α - cut terval are gve below: ower level α - cut terval: ake Sample (Observato j) α 4 + α 6 + α 8 + α 5 + α B 6 + α 8 + α 9 + α 9 + 3α + α 4 + α + α + 3α + 3α + α 97

12 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg The ull hypothess : μ μ μ agast the alteratve hypothess 3 :μ μ μ 3 pper level α - cut terval: ake Sample (Observato j) α - α - α 4 - α - 3α B 3-3α - α 5 - α 5 - α 9-3α 9 - α 7 - α 9 - α - α 7-3α The ull hypothess : μ μ μ agast the alteratve hypothess 3 :μ μ μ 3 The NOV table for lower level model: ere, N = 5 ad SV SS df S F-rato F Betwee lasses Wth lasses h - = 3 N - h = 5 3 5, 5, 5 for the makes, B, respectvely F T = 7 + 3α ; T 3α + 44α ad 5 j a j+ α bj- aj 69α + 9α ; α - 76α ad 74α - α α + 8α d α - 76α ; 4α + 8α ad α - 76α + 88 F α + 9α + 36 where α ad F s the calculated value of F at lower level model Now, the tabulated value of F at k = 5% level of sgfcace wth h -, N - h, degrees of freedom s F 388 F > F α, α, we reject the ull hypothess t at 5% Sce, t at 5% There s a sgfcat dfferece the durablty of the 3 makes of computers at lower level of α - cut 98

13 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg The NOV table for upper level model: ere, N = 5 ad SV SS df S F-rato F Betwee lasses Wth lasses h - = 3 N - h = 5 3 5, 5, 5 for the makes, B, respectvely F T = 6-8α ; T 64α - 3α ad 5 j dj- α dj- cj 6α - 596α ; 8α + 6α ad 46α + 76α α + α d 8α + 6α ; 46α + α ad 8α + 6α F 3α + α + 3 where α ad F s the calculated value of F at upper level model d the tabulated value of F at k = 5% level of sgfcace wth h -, N - h, degrees of freedom s F 388 F > F α, α, we reject the ull hypothess t at 5% ere, t at 5% There s a sgfcat dfferece the durablty of the 3 makes of computers at upper level of α - cut ocluso 4 Therefore, the ull hypotheses ad are rejected α, α s a sgfcat dfferece betwee the durablty of the 3 makes of computers We coclude geeral that there Refereces [] S bbasbady, B sady, The earest trapezodal fuzzy umber to a fuzzy quatty, ppl ath omput 56 (4) [] S bbasbady, mrfakhra, The earest approxmato of a fuzzy quatty parametrc form, ppl ath omput 7 (6) [3] S bbasbady, mrfakhra, The earest trapezodal form of geeralzed left rght fuzzy umber, Iterat J pprox Reaso 43 (6) [4] S bbasbady, T ajjar, Weghted trapezodal approxmato-preservg cores of a fuzzy umber, omputers ad athematcs wth pplcatos 59 () [5] ref ad S Taher, Testg fuzzy hypotheses usg fuzzy data based o fuzzy test statstc, Joural of certa Systems, Vol 5,, 45-6 [6] B F rold, Testg fuzzy hypotheses wth crsp data, Fuzzy Sets ad Systems, Vol 94, 998,

14 athematcal Theory ad odelg ISSN (Paper) ISSN 5-5 (Ole) Vol5, No3, 5 wwwsteorg [7] S Bodjaova, eda value ad meda terval of a fuzzy umber, Iform Sc 7 (5) [8] J J Buckley, Fuzzy statstcs, Sprger-Verlag, New York, 5 [9] J hach, S Taher ad R Vertl, Testg statstcal hypotheses based o fuzzy cofdece tervals, Forschugsbercht S--, Techsche verstat We, ustra, [] S haas, O the terval approxmato of a fuzzy umber, Fuzzy Sets ad Systems (999) -6 [] S he ad G, Represetato, rakg, ad dstace of fuzzy umber wth expoetal membershp fucto usg graded mea tegrato method, Tamsu, Oxf J ath Sc 6 () () 3-3 [] D Dubos ad Prade, Operatos o fuzzy umbers, It J Syst Sc 9 (978) [3] D Dubos ad Prade, Fuzzy sets ad systems: Theory ad applcato, cademc Press, New York, 98 [4] P Grzegorzewsk, Testg statstcal hypotheses wth vague data, Fuzzy Sets ad Systems, Vol,, 5-5 [5] P Grzgorzewsk, E rowka, Trapezodal approxmato of fuzzy umbers, Fuzzy Sets ad Systems 53 (5) 5-35 [6] S Gupta, V K Kapoor, Fudametals of mathematcal statstcs, Sulta had & Sos, New Delh, Ida [7] R R ockg, ethods ad applcatos of lear models: regresso ad the aalyss of varace, New York: Joh Wley & Sos, 996 [8] D Kalpaaprya, P Pada, Fuzzy hypothess testg of NOV model wth fuzzy data, It Nat J oder Egg Research, Vol, Issue 4 July-ug () [9] George J Klr ad Bo Yua, Fuzzy sets ad fuzzy logc, Theory ad pplcatos, Pretce-all, New Jersey, 8 [] a, Kadel, Fredma, ew approach for defuzzfcato, Fuzy Sets ad Systems () [] K G ato, Woodbury ad D Tolley, Statstcal applcatos usg fuzzy sets, New York: Joh Wley & Sos, 994 [] khko Kosh, Tetsuj Okuda ad Kyoj sa, alyss of varace based o fuzzy terval data usg momet correcto method, Iteratoal Joural of Iovatve omputg, Iformato ad otrol, Vol, 6, [3] J J Saade ad Schwarzlader, Fuzzy hypothess testg wth hybrd data, Fuzzy Sets ad Systems, Vol 35, 99, 97- [4] J J Saade, Exteso of fuzzy hypothess testg wth hybrd data, Fuzzy Sets ad Systems, Vol 63, 994, 57-7 [5] T Veeraraja, Probablty, statstcs ad radom process, Tata cgraw ll Educato Pvt td, New Delh, Ida [6] R Vertl, varate statstcal aalyss wth fuzzy data, omputatoal Statstcs ad Data alyss, Vol 5, 6, [7] R Vertl, Statstcal methods for fuzzy data, Joh Wley ad Sos, hchester, [8] N Wataabe ad T Imazum, fuzzy statstcal test of fuzzy hypotheses, Fuzzy Sets ad Systems, Vol 53, 993, [9] Wu, alyss of varace for fuzzy data, Iteratoal Joural of Systems Scece, Vol 38, 7, [3] Wu, Statstcal cofdece tervals for fuzzy data, Expert Systems wth pplcatos, Vol 36, 9, [3] Wu, Statstcal hypotheses testg for fuzzy data, Iformato Sceces, Vol 75, 5, 3-56 [3] Zadeh, Fuzzy sets, Iformato ad otrol 8 (965)