A Comparative Study of Chi-Square Goodness-of-Fit Under Fuzzy Environments

Transcription

1 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 Comparative Study of Chi-Square Goodess-of-Fit der Fuzzy Eviromets S Parthiba 1 ad P Gajivaradha 1 Research Scholar, Departmet of Mathematics, Pachaiyappa s College, Cheai-6 3, Tamil Nadu, Idia Departmet of Mathematics, Pachaiyappa s College, Cheai-6 3, Tamil Nadu, Idia bstract Testig goodess-of-fit plays a vital role i data aalysis This problem seems to be much more complicated i the presece of vague data I this paper, the chi-square goodess-of-fit uder trapezoidal fuzzy umbers (tfs) is proposed usig alpha cut iterval method d the rakig grades of tfs are also used to compute the chisquare test statistic The proposed techique is illustrated with two differet umerical examples alog with differet methods of rakig grades for a cocrete comparative study Keywords: Chi-square Test, Fuzzy Sets, Trapezoidal Fuzzy Numbers, lpha Cut, Rakig Fuctio, Graded Mea Itegratio Represetatio 1 Itroductio: Most of statistical procedures are based o fairly specific assumptios regardig the uderlyig populatio distributio, like ormality, expoetiality, etc Therefore it might be desirable to check whether these assumptios are reasoable Statistical procedures for testig hypotheses about the uderlyig distributio are called goodess-of-fit test Fuzzy set theory [34] has bee applied to may areas which eed to maage ucertai ad vague data Such areas iclude approximate reasoig, decisio makig, optimizatio, cotrol ad so o I traditioal statistical testig [13], the observatios of sample are crisp ad a statistical test leads to the biary decisio owever, i the real life, the data sometimes caot be recorded or collected precisely The statistical hypotheses testig uder fuzzy eviromets has bee studied by may authors usig the fuzzy set theory cocepts itroduced by Zadeh [34] Viertl [9] ivestigated some methods to costruct cofidece itervals ad statistical tests for fuzzy data Wu [3] proposed some approaches to costruct fuzzy cofidece itervals for the ukow fuzzy parameter ew approach to the problem of testig statistical hypotheses is itroduced by Chachi et al [8] Mikihiko Koishi et al [19] proposed a method of NOV for the fuzzy iterval data by usig the cocept of fuzzy sets ypothesis testig of oe factor NOV model for fuzzy data was proposed by Wu [31, 33] usig the h-level set ad the otios of pessimistic degree ad optimistic degree by solvig optimizatio problems Gajivaradha ad Parthiba aalysed oe-way NOV test usig alpha cut iterval method for trapezoidal fuzzy umbers [] ad they preseted a comparative study of -factor NOV test uder fuzzy eviromets usig various methods [1] Wag et al preseted a method for cetroid formulae for a geeralized fuzzy umber ad arrived some differet approach for rakig tfs [3] Salim Rezvai aalysed the rakig fuctios with tfs [5] Thorai et al approached the rakig fuctio of a tfs with some modificatios [6] Salim Rezvai ad Mohammad Molai preseted the shape fuctio ad Graded Mea Itegratio Represetatio for tfs [4] I this paper, we propose the chi-square goodess-of-fit uder fuzzy data That is, if the observed large samples are uavoidably i terms of trapezoidal fuzzy umbers (or triagular fuzzy umbers), we suggest here how to modify the classical chi-square test for such data usig their alpha cut itervals d the decisio rules of the proposed techique are give I the proposed approach, the degrees of optimism, pessimism ad h-level set are ot used but used i Wu [31] I fact we would like to preset a coclusio that α-cut iterval method is geeral eough to deal with chi-square test of goodess-of-fit uder fuzzy data (tfs) lso, i this paper we have aalysed what ca be the result if the cetroid/rakig grades of tfs are employed i hypotheses testig The same cocept ca also be applied for the data which are i terms of triagular fuzzy umbers For better uderstadig, the proposed techique is illustrated with two differet kids of umerical examples with differet coclusios 8

2 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 Prelimiaries Defiitio 1 Geeralized fuzzy umber geeralized fuzzy umber is described as ay fuzzy subset of the real lie fuctio μ x satisfies the followig coditios: i μ x is a cotiuous mappig from to the closed iterval ii μ x =, for all x -, a, iii μ x x is strictly icreasig o a, b, iv μ x ω, for all b, c, as ω is a costat ad < ω 1, v μr x R x is strictly decreasig o c, d, vi μ x, for all x d,, ω, ω 1,, whose membership where a, b, c, d are real umbers such that a < b c < d Defiitio fuzzy set is called ormal fuzzy set if there exists a elemet (member) x such that μ x 1 μ αx α x mi μ x, μ x fuzzy set is called covex fuzzy set if 1 1 where x, x X ad α, 1 The set α x X μ x α set 1 is said to be the α - cut of a fuzzy Defiitio 3 fuzzy subset of the real lie with membership fuctio μ μ x :, 1, is called a fuzzy umber if is ormal, is fuzzy covex, μ Supp is bouded, where Supp clx : μ x x is upper semicotiuous ad operator x such that ad cl is the closure Defiitio 4 α-cut of a fuzzy umber: useful otio for dealig with a fuzzy umber is a set of its α-cuts The α-cut of a fuzzy umber is a o-fuzzy set defied as α={x :μ (x) α} family of { : α (,1]} is a set represetatio of the fuzzy umber ccordig to the defiitio of a α fuzzy umber, it is easily see that every α-cut of a fuzzy umber is a closed iterval ece we have, [, ] where if{x :μ (x) α} ad sup{x :μ (x) α} space of α α α all fuzzy umbers will be deoted by F( ) α It is kow that for a ormalized tf (a, b, c, d; 1), there exists four umbers a, b, c, d ad two fuctios x, R x :, 1, where x ad R x are o-decreasig ad oicreasig fuctios respectively d its membership fuctio is defied as follows: μ x x =(x-a)/(b-a) for a x b; 1 for b x c; otherwise The fuctios x ad R respectively [9] I this paper, we assume that umber are strictly mootoe, obviously, α R x =(x-d)/(c-d) for c x d ad x are also called the left ad right side of the fuzzy umber x dx < + The left ad right sides of the fuzzy ad are iverse fuctios of respectively other importat type of fuzzy umber was itroduced i [6] as follows: et a, b, c, d such that a < b c < d x ad R x fuzzy umber defied as μ x :, 1, 83

3 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 x - a d - x μ x for a x b; 1 for b x c; for c x d; b - a d - c deoted by a, b, c, d d x right spread of the tf [Dubois ad Prade i 1981] a, b, c, d x - a b - a ; d - x R x d - c If, the[1-4], α α, α a + b - a α, d - d - c α ; α, 1 otherwise where > is ca also be termed as left ad Whe = 1 ad b = c, we get a triagular fuzzy umber The coditios r = 1, a = b ad c = d imply the closed iterval ad i the case r = 1, a = b = c = d = t (some costat), we ca get a crisp umber t Sice a trapezoidal fuzzy umber is completely characterized by = 1 ad four real umbers a b c d, it is T a, b, c, d F ofte deoted as d the family of trapezoidal fuzzy umbers will be deoted by for = 1we have a ormal trapezoidal fuzzy umber a, b, c, d ad the correspodig α - cut is defied by α a + αb - a, d - αd - c ; α, 1 (5) d we eed the followig results which ca be foud i [13, 14] Result 1 et D = {[a, b], a b ad a, b }, the set of all closed, bouded itervals o the real lie Result et = [a, b] ad B = [c, d] i D The = B if a = c ad b = d 3 Chi-square distributio If X i (,,, ) are idepedet ormal variates with mea the χ (X -μ ) / σ i i i μi ad variace σ (,,, ) is a chi-square variate with degrees of freedom The probability desity ( ) 1 χ fuctio of the chi-square distributio is give by, f(χ )=(1/( ))(χ ) e ; χ where is the degrees of freedom d the exact shape of the distributio depeds upo the umber of degrees of freedom I geeral, whe is small, the shape of the curve is skewed to the right ad as υ gets larger, the distributio becomes more ad more symmetrical The mea ad variace of the chi-square distributio are ad respectively s, the chi-square distributio approaches a ormal distributio The sum of idepedet chi-square variates is also a chi-square variate Moreover, chi-square distributio is very useful: (i) to test if the hypothetical value of the populatio variace is σ =σ (say) (ii) to test the goodess of fit It is used to determie whether a actual sample distributio matches a kow theoretical distributio (iii) to test the idepedece of attributes ie if a populatio is kow to have two attributes, the chi-square distributio is used to test whether the two attributes are associated or idepedet, based o a sample (iv) to test the homogeeity of idepedet estimates of the populatio correlatio coefficiet 31 Coditios for the validity of chi-square test: (i) The experimetal data (sample observatios) must be idepedet of each other (ii) The total frequecy (or umber of observatios i the sample) must be reasoably large, say 5 (iii) No idividual frequecies should be less tha 5, if ay frequecy is less tha 5, the it is pooled with the precedig or succeedig frequecy so that the pooled frequecy is more tha 5 Fially adjust for the degrees of freedom lost i poolig (iv) The umber of classes must be either too small or too large ie

4 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 3 Chi-square test of goodess of fit: Tests of goodess of fit [1, 7] are used whe we wat to determie whether a actual sample distributio matches a kow theoretical distributio It eables us to fid if the deviatio of the experimet from theory is just by chace or it is really due to the iadequacy of the theory to fit the observed data If, (,, ) is a set of observed frequecies ad E i, (,, ) is the correspodig set of expected frequecies, the i i i χ ((O - E ) /E ) follows chi-square distributio with (-1) degrees of freedom Suppose that a radom sample X 1,, X is draw from a populatio with ukow cumulative distributio fuctio F We wish to test the ull hypothesis :F(x) = F (x) x that the populatio cdf is F (which is completely specified), agaist :F(x) F (x) for some x To apply this test, the data must first be grouped ito categories ad the the observed frequecies for these categories are compared with the frequecies expected uder the ull hypothesis I the case of a discrete distributio these categories appear i a atural way ad are relevat to the distributio uder study Whe the distributio F is cotiuous we have to arrage classes which are couterparts of above metioed categories et k be the level of sigificace If the calculated χ χ k with (-1) degrees of freedom, we will accept the ull hypothesis the the differece betwee the observed ad expected frequecies is ot sigificat at k% level of sigificace If χ χ k, we reject ad coclude that the differece is sigificat It may happe that a sample used for makig decisio cosists of observatios that are ot ecessarily crisp but may be vague as well I order to describe the vagueess of data we use the otio of a fuzzy umber, itroduced by Dubois ad Prade [9] 33 Fuzzy radom variables: otio of fuzzy radom variables was itroduced by Kwakeraak [17, 18] Other defiitios of fuzzy radom variables are due to Kruse [15] or to Puri ad Ralescu [] Suppose that a radom experimet is described as usual by a probability space (,, ), where is the set of all possible outcomes of the experimet, is the σ -algebra of subsets of ( the set of all possible evets) ad is a probability measure The the mappig X : F( ) is called a fuzzy radom variable if {X(α, ω): α (, 1]} is a set represetatio of X(ω) for all ω ad for each α (, 1] both X X (ω) sup X (ω) are usual real-valued radom variables[16] o (,, ) α α α X X (ω) if X (ω) ad α α α fuzzy radom variable X is cosidered as a perceptio of a ukow usual radom variable V:, called a origial of X (if oly vague data are available, it is of course impossible to show which of the possible origials is the true oe) Similarly -dimesioal fuzzy radom sample X 1,, X may be treated as a fuzzy perceptio [16] of the usual radom sample V 1,, V (where V 1,, V are idepedet ad idetically distributed crisp radom variables) radom variable is completely characterized by its probability distributio P I statistical reasoig we assume that a probability distributio uder study belogs to a family of θ distributios = { P :θ} where is the parameter space The very ofte we idetify the distributio with θ its parameter θ ad restrict statistical iferece to that parameter owever, if we deal with a fuzzy radom variable, we caot observe the parameter θ directly but oly its vague image sig this reasoig together with Zadeh s extesio priciple Kruse ad Meyer [16] itroduced the otio of fuzzy parameter of fuzzy radom variable θ which may be cosidered as a fuzzy perceptio of the ukow parameter θ It is defied as a fuzzy subset of the parameter space with membership fuctio μ : [, 1] Of course, if our data are crisp ie X = V, we get θ θ 34 Chi-square test for vague data: Suppose μ,, μ deote membership fuctios of fuzzy umbers which are observatios of a X1 X fuzzy radom sample X 1,,X Suppose our sample comes from the ukow distributio F, ad our aim is 85

5 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 to test the ull hypothesis : F F θ agaist : F F θ where the distributio a fuzzy parameter θ described by its membership fuctio with fuzzy data is give by [1, 11] F is completely specified by θ μ d the test statistic for testig agaist χ ((Oi-E i) /E i) (35) for large samples ad which follows approximately chi-square distributio with (-1) degrees of freedom Therefore, we reject i favor of if χ χ 1-α, -1 degrees of freedom where χ is the quatile of order 1-α from the chi-square distributio with k-1 1-α, -1 4 Chi-square goodess of fit usig alpha cut iterval method: The fuzzy test of hypotheses of chi-square model i which the sample data are trapezoidal fuzzy umbers is give here The test statistic for fuzzy observatios give by (35) is formulated accordig to the α-cut iterval of tfs (def 4; sectio ) sig the relatio (def 5; sectio ), we trasform the fuzzy chisquare model to iterval chi-square model avig the upper limit of the alpha cut iterval, we costruct upper level crisp chi-square model ad usig the lower limit of the alpha cut iterval, we costruct the lower level crisp chi-square model Thus, i this approach, the test statistic (35) is split ito two parts up amely lower level ad upper level α-cut itervals viz (α)=[a i + α(bi - a i)]---(41) ad (α)=[d i- α(di - c i)]---(4) ;,, ; α [, 1] ccordigly, the test statistics will be - E i - Ei ad χ (43) Ei i i i i ad O [d - α(d - c )];,, ; α [, 1] χ (44) Ei where i i i i O [a + α(b - a )] Decisio rules: The decisio rules for the fuzzy hypotheses are give below: : F F θ agaist : F F θ : F F θ agaist : F F θ, : F,F F, F θ θ agaist, : F,F F, F θ θ The ull hypothesis for lower level model: : F F θ agaist : F F θ The ull hypothesis for upper level model: : F F θ agaist : F F θ Example 1 The followig table shows defective articles produced by four machies i,,, 3, 4 Due to some work cogestio, the observed data are uavoidably trapezoidal fuzzy umbers Machie Productio time (i hours) Number of defectives (1, 1, 13, 15) (6, 9, 3, 3) (58, 6, 63, 64) (94, 96, 98, 11) We ow test whether the tfs idicate a sigificat differece i the performace of the machies Example The demad for a particular spare part i a factory was foud to vary from day-to-day The observed demad of the spare parts are i terms of tfs due to some uexpected situatios i the o-stop work flow The obtaied sample study is tabulated below: 86

6 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 Days Mo Tues Wed Thurs Fri Sat No of parts (1119, 11, (11, 11, (117, 118, (1116, 11, (111, 114, (1114, 1115, demaded 114, 116) 115, 118) 111, 1114) 11, 113) 116, 117) 1117, 11) We test the hypotheses whether the umber of parts demaded depeds o the day of the week Example 41 et us cosider example 1, usig the relatio (5) i sectio, the alpha cut iterval model for the 4 machies are give by, Machie Productio time (i hours) Number of defectives [1+, 15-] [6+3, 3-] [58+, 64-] [94+, 11-3] : Productio rates of the 4 machies are same The lower level model (llm) Machie Productio time (i hours) Number of defectives [1+] [6+3] [58+] [94+] : Productio rates of the 4 machies are same The total umber of defectives at llm = [188+9 The expected umber of defectives ad observed umber of defectives produced by the four machies are give below respectively, E i (1/7)[188+9] (1/7)[188+9] (/7)[188+9] (3/7)[188+9] [1+] [6+3] [58+] [94+] - Ei the test statistic for llm is χ 14α -1355α+1413 Ei 378α+7896 d sice Ei Oi, v =4-1=3, from the chi-square table, χ ( v 3) 7815 ere, χ > χ α, α 1 The ull hypothesis is rejected at 5% level of sigificace α There is a sigificat differece i the performace of machies at llm The upper level model (ulm) Machie Productio time (i hours) Number of defectives [15-] [3-] [64-] [11-3] : Productio rates of the 4 machies are same The total umber of defectives at ulm = [1-8 87

7 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 The expected umber of defectives ad observed umber of defectives produced by the four machies are give below respectively, E i (1/7)[1-8] (1/7)[1-8] (/7)[1-8] (3/7)[1-8] [15-] [3-] [64-] [11-3] - Ei the test statistic for ulm is χ i E 693α +8988α α d sice Ei Oi, v =4-1=3, from the chi-square table, χ ( v 3) 7815 ere, χ > χ α, α 1 The ull hypothesis There is a sigificat differece i the performace of machies at ulm is rejected at 5% level of sigificace α ece, observig the decisios from both llm ad ulm, the ull hypothesis is rejected at 5% level of sigificace ad we coclude that there is a sigificat differece i the performace of 4 machies Example 4 et us cosider example, usig the relatio (5) i sectio, the alpha cut iterval of the give tfs are tabulated below: Days Mo Tues Wed Thurs Fri Sat [1119+3, [11+, [117+, [1116+4, [111+3, [1114+, Demad 116-] 118-3] ] 113-] 117-] 11-3] : The umber of spare parts demaded are same over the 6-day The lower level model (llm) Days Mo Tues Wed Thurs Fri Sat Demad [1119+3] [11+] [117+] [1116+4] [111+3] [1114+] : The umber of spare parts demaded are same over the 6-day period The total umber of defectives at llm = [ der the ull hypothesis, the expected frequecies of the spare parts demaded o each of the six days would be [6697 E i [6697 [6697 [6697 [6697 [6697 [6697 [1119+3] [11+] [117+] [1116+4] [111+3] [1114+] - Ei the test statistic for llm is χ 64α +1344α+4854 Ei 84α+418 d sice Ei Oi, v =6-1=5, from the chi-square table, χ ( v 5) 117 ere, χ < χ α, α 1The ull hypothesis is accepted at 5% level of sigificace α 88

8 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 The umber of spare parts demaded are same over the 6-day period at llm The upper level model (ulm) Days Mo Tues Wed Thurs Fri Sat Demad [116-] [118-3] [1114-4] [113-] [117-] [11-3] : The umber of spare parts demaded are same over the 6-day period The total umber of defectives at llm = [ der the ull hypothesis, the expected frequecies of the spare parts demaded o each of the six days would be [6738 i [6738 [6738 [6738 [6738 [6738 [6738 E [116-] [118-3] [1114-4] [113-] [117-] [11-3] - Ei the test statistic for llm is χ 1α +7α+5 Ei 14-4α d sice Ei Oi, v =6-1=5, from the chi-square table, χ ( v 5) 117 ere, χ < χ α, α 1The ull hypothesis is accepted at 5% level of sigificace α The umber of spare parts demaded are same over the 6-day period at ulm ece, observig the decisios obtaied from both llm ad ulm, the ull hypothesis is accepted at 5% level of sigificace ad we coclude that the umber of spare parts demaded are same over the 6- day period 5 Wag s cetroid poit ad rakig method Wag et al [3] foud that the cetroid formulae proposed by Cheg are icorrect ad have led to some misapplicatios such as by Chu ad Tsao They preseted the correct method for cetroid formulae for a geeralized fuzzy umber = a, b, c, d; w as 1 dc - ab w c - b x, y a + b + c + d, (51) 3 d + c-a + b 3 d + c-a + b d the rakig fuctio associated with is R x + y --- (5) For a ormalized tf, we put w = 1 i equatios (51) so we have, 1 dc - ab 1 c - b x, y a + b + c + d, (53) 3 d + c-a + b 3 d + c-a + b d the rakig fuctio associated with is R x + y --- (54) et i ad j be two fuzzy umbers (ii) R i < R j the i j (iii) R i = R j the i j (i) R R the i j i j 89

9 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 Example 51 et us cosider example 1, the rakig grades of tfs are calculated usig relatios (53) ad (54) which are give below: d total o of defectives =, whole umbers subject to the coditio that see that χ ((Oi-E i) /E i) 18 E i : /7; /7; ()/7; 3()/7 ad covertig E i to the Ei =, we get,, ; sice Ei Oi Machie Productio time (i hours) Number of defectives E i χ ( v 3) 7815 ere χ > χ The ull hypothesis v, from the chi-square table it is sigificace The differece betwee the performaces of 4 machies is sigificat is rejected at 5% level of Example 5 et us cosider example, the rakig grades of tfs are calculated usig relatios (53) ad (54) which are give below: Days Mo Tues Wed Thurs Fri Sat Demad d total o of demad = , is see that χ ((Oi-E i) /E i) 1376 E i = (total o of defectives/6) = (671749/6) = , ; sice Ei Oi χ ( v 5) 117 ere χ < χ The ull hypothesis E i v, from the chi-square table it sigificace The umber of spare parts demaded are same over the 6-day period is accepted at 5% level of 6 Rezvai s rakig fuctio of TFNs The cetroid of a trapezoid is cosidered as the balacig poit of the trapezoid Divide the trapezoid ito three plae figures These three plae figures are a triagle (PB), a rectagle (BPQC) ad a triagle (CQD) respectively et the cetroids of the three plae figures be 1 3 G, G ad G respectively The iceter of these cetroids G 1, G ad G 3 is take as the poit of referece to defie the rakig of geeralized trapezoidal fuzzy umbers The reaso for selectig this poit as a poit of referece is that each cetroid poit are balacig poits of each idividual plae figure ad the iceter of these cetroid poits is much more balacig poit for a geeralized trapezoidal fuzzy umber Therefore, this poit would be a better referece poit tha the cetroid poit of the trapezoid 9

10 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 Cosider a geeralized trapezoidal fuzzy umber = a, b, c, d; w The cetroids of the three plae figures are: a+b w b+c w c+d w G 1,, G, ad G 3, --- (61) Equatio of the lie GG 1 3is w y = ad G does ot lie o the lie GG 1 3 Therefore, G 1, G ad G 3 3 are o-colliear ad they form a triagle We defie the iceter Ix, y of the triagle with vertices G 1, G ad G 3 of the geeralized fuzzy umber = a, b, c, d; w as [5], a+b b+c c+d w w w α β γ α β γ I x, y, --- (6) α + β + γ α + β + γ c - 3b + d w c + d - a - b 3c - a - b w where α,β,γ --- (63) = a, b, c, d; w which maps the set of all fuzzy d rakig fuctio of the trapezoidal fuzzy umber umbers to a set of all real umbers ie R: is defied as R 91 x + y --- (64) which is the Euclidea distace from the iceter of the cetroids For a ormalized tf, we put w = 1 i equatios (61), (6) ad (63) so we have, a+b 1 b+c 1 c+d 1 G 1,, G, ad G 3, --- (65) a+b b+c c+d α β γ α β γ I x, y, --- (66) α + β + γ α + β + γ c - 3b + d 1 c + d - a - b 3c - a - b 1 where α,β ad γ --- (67) d rakig fuctio of the trapezoidal fuzzy umber (68) = a, b, c, d; 1 is defied as R x + y ---

11 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 7 Chi-square test usig Rezvai s rakig fuctio We ow aalyse the chi-square test by assigig rak for each ormalized trapezoidal fuzzy umbers ad based o the rakig grades the decisios are observed Example 71 et us cosider example 1, the rakig grades of tfs are calculated usig the relatios (66), (67) ad (68) which are give below: d total o of defectives =, defectives are tabulated below: see that χ ((Oi-E i) /E i) 134 E i : /7; /7; ()/7; 3()/7, the expected ad observed umber of, ; sice Ei Oi χ ( v 3) 7815 ere χ > χ The ull hypothesis v, from the chi-square table it is sigificace The differece betwee the performaces of 4 machies is sigificat is rejected at 5% level of Example 7 et us cosider example, the rakig grades of tfs are calculated usig relatios (66), (67) ad (68) which are give below: d total o of demad = 67175, is see that χ ((Oi-E i) /E i) 1636 E i = (total o of defectives/6) = (67175/6) = , ; sice Ei Oi χ ( v 5) 117 ere χ < χ The ull hypothesis v, from the chi-square table it sigificace The umber of spare parts demaded are same over the 6-day period is accepted at 5% level of 8 Thorai s cetroid poit ad rakig method s per the descriptio i Salim Rezvai s rakig method, Y P Thorai et al [6] preseted a differet kid of cetroid poit ad rakig fuctio of tfs The iceter G, G ad G of the geeralized tf 1 3 Machie Productio time (i hours) Number of defectives E i Days Mo Tues Wed Thurs Fri Sat Demad E i = a, b, c, d; w is give by, I x, y of the triagle [Fig 1] with vertices a+b b+c c+d w w w α β γ α β γ I x, y, --- (81) α + β + γ α + β + γ c - 3b + d w c + d - a - b 3c - a - b w where α,β,γ --- (8)

12 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 d the rakig fuctio of the geeralized tf = a, b, c, d; w which maps the set of all fuzzy umbers to a set of real umbers is defied as R xy--- (83) For a ormalized tf, we put w = 1 i equatios (81) ad (8) so we have, a+b b+c c+d α β γ α β γ I x, y, --- (84) α + β + γ α + β + γ c - 3b + d 1 c + d - a - b 3c - a - b 1 where α,β ad γ --- (85) = a, b, c, d; 1, R x y d for --- (86) 9 Chi-square test usig Thorai s rakig fuctio We ow aalyse the chi-square test by assigig rak for each ormalized trapezoidal fuzzy umbers ad based o the rakig grades the decisios are observed Example 91 et us cosider example 1, the rakig grades of tfs are calculated usig the relatios (84), (85) ad (86) which are give below: d total o of defectives = 83, defectives are tabulated below: see that χ ((Oi-E i) /E i) 474 E i : 83/7; 83/7; (83)/7; 3(83)/7, the expected ad observed umber of, ; sice Ei Oi Machie Productio time (i hours) Number of defectives E i χ ( v 3) 7815 ere χ < χ The ull hypothesis v, from the chi-square table it is sigificace The differece betwee the performaces of 4 machies is ot sigificat is accepted at 5% level of Example 9 et us cosider example, the rakig grades of tfs are calculated usig relatios (84), (85) ad (86) which are give below: Days Mo Tues Wed Thurs Fri Sat Demad d total o of demad = , is see that χ ((Oi-E i) /E i) 687 E i = (total o of defectives/6) = (797748/6) = , ; sice Ei Oi χ ( v 5) 117 ere χ < χ The ull hypothesis E i v, from the chi-square table it sigificace The umber of spare parts demaded are same over the 6-day period is accepted at 5% level of 93

13 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 1 Graded mea itegratio represetatio (GMIR) et = a, b, c, d; w be a geeralized trapezoidal fuzzy umber, the the GMIR [4] of is defied by w -1-1 w h R h P h dh / hdh Theorem 11 et = a, b, c, d; 1 be a tf with ormal shape fuctio, where a, b, c, d are real umbers such that a < b c < d The the graded mea itegratio represetatio (GMIR) of is a + d P b - a - d + c + 1 Proof : For a trapezoidal fuzzy umber = a, b, c, d; 1, we have x R x d - x d - c The, b - a x - a h = h a + b - a h 1-1 d - x 1-1 h = R h d -d - ch d - c P h a + b - ah d -d - ch dh / hdh 1 a + d = b - a - d + c / a + d Thus, P b - a - d + c ece the proof + 1 a + b + c + d Result 11 If =1 i the above theorem, we have P 6 ; x - a b - a 11 Chi-square test usig GMIR of tfs We ow aalyse the chi-square test by usig GMIR of each ormalized trapezoidal fuzzy umbers ad based o the GMIR of tfs the decisios are observed Example 111 et us cosider example 1, the GMIRs of tfs are calculated usig the result (11) of theorem 11 which are give below: d total o of defectives =, whole umbers subject to the coditio that see that χ ((Oi-E i) /E i) 18 E i : /7; /7; ()/7; 3()/7 ad covertig E i to the Ei =, we get,, ; sice Ei Oi Machie Productio time (i hours) Number of defectives E i χ ( v 3) 7815 ere χ > χ The ull hypothesis ad v, from the chi-square table it is sigificace The differece betwee the performaces of 4 machies is sigificat is rejected at 5% level of 94

14 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 Example 11 et us cosider example, the GMIRs of tfs are calculated usig the result (11) of theorem 11 which are give below: Days Mo Tues Wed Thurs Fri Sat Demad d total o of demad = 67175, is see that χ ((Oi-E i) /E i) 1485 E i = (total o of defectives/6) = (67175/6) = E i , ; sice Ei Oi χ ( v 5) 117 ere χ < χ The ull hypothesis v, from the chi-square table it sigificace The umber of spare parts demaded are same over the 6-day period 1 Coclusio The decisios obtaied from various methods are tabulated below for the ull hypothesis cceptace of ull hypotheses cut method Wag Rezvai Thorai GMIR is accepted at 5% level of Eg1 Eg Eg1 Eg Eg1 Eg Eg1 Eg Eg1 Eg ere, the proposed α -cut iterval method provides a parallel decisio for the acceptace/rejectio of ull hypothesis i lower level () ad upper level () models for both example 1 ad example Wag s rakig method, Rezvai s rakig method ad GMIR of tfs exhibit the same decisios for example 1 ad example Thorai s rakig method of tfs does ot provide reliable result as it accepts the ull hypothesis i all the cases REFERENCES 1 S bbasbady, B sady, The earest trapezoidal fuzzy umber to a fuzzy quatity, ppl Math Comput, 156 (4) S bbasbady, M mirfakhria, The earest approximatio of a fuzzy quatity i parametric form, ppl Math Comput, 17 (6) S bbasbady, M mirfakhria, The earest trapezoidal form of geeralized left right fuzzy umber, Iterat J pprox Reaso, 43 (6) S bbasbady, T ajjari, Weighted trapezoidal approximatio-preservig cores of a fuzzy umber, Computers ad Mathematics with pplicatios, 59 (1) E Baloui Jamkhaeh ad Nadi Ghara, Testig statistical hypotheses for compare meas with vague data, Iteratioal Mathematical Forum, 5 (1) S Bodjaova, Media value ad media iterval of a fuzzy umber, Iform Sci 17 (5) J J Buckley, Fuzzy statistics, Spriger-Verlag, New York, 5 8 J Chachi, S M Taheri ad R Viertl, Testig statistical hypotheses based o fuzzy cofidece itervals, Forschugsbericht SM-1-, Techische iversitat Wie, ustria, 1 9 D Dubois ad Prade, Operatios o fuzzy umbers, It J Syst Sci, 9 (1978) P Grzegorzewski, Statistical iferece about the media from vague data, Cotrol ad Cyberetics 7 (1998), P Grzegorzewski, Testig statistical hypotheses with vague data, Fuzzy Sets ad Systems 11 (),

15 Mathematical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol6, No, 16 1 S C Gupta, V K Kapoor, Fudametals of mathematical statistics, Sulta Chad & Sos, New Delhi, Idia 13 R R ockig, Methods ad applicatios of liear models: regressio ad the aalysis of variace, New York: Joh Wiley & Sos, George J Klir ad Bo Yua, Fuzzy sets ad fuzzy logic, Theory ad pplicatios, Pretice-all, New Jersey, 8 15 R Kruse, The strog law of large umbers for fuzzy radom variables, Iform Sci 8 (198), R Kruse, K D Meyer, Statistics with Vague Data, D Riedel Publishig Compay, Kwakeraak, Fuzzy radom variables, part I: Defiitios ad theorems, Iform Sci 15 (1978), Kwakeraak, Fuzzy radom variables, part II: lgorithms ad examples for the discrete case, Iform Sci 17 (1979), Mikihiko Koishi, Tetsuji Okuda ad Kiyoji sai, alysis of variace based o Fuzzy iterval data usig momet correctio method, Iteratioal Joural of Iovative Computig, Iformatio ad Cotrol, (6) S Parthiba, ad P Gajivaradha, Oe-factor NOV model usig trapezoidal fuzzy umbers through alpha cut iterval method, als of Pure ad pplied Mathematics, 11(1) (16) S Parthiba, ad P Gajivaradha, comparative study of two factor NOV model uder fuzzy eviromets usig trapezoidal fuzzy umbers, It J of Fuzzy Mathematical rchive, 1(1) (16) 1-5 M Puri, D Ralescu, Fuzzy radom variables, J Math al ppl 114 (1986), S Salahsour, S bbasbady ad T llahviraloo, Rakig Fuzzy Numbers usig Fuzzy Maximizig- Miimizig poits, ESFT-F: July 11, ix-les-bais, Frace 4 Salim Rezvai ad Mohammad Molai, Represetatio of trapezoidal fuzzy umbers with shape fuctio, to appear i als of Fuzzy mathematics ad Iformatics 5 Salim Rezvai, Rakig Geeralized Trapezoidal Fuzzy Numbers with Euclidea Distace by the Icetre of Cetroids, Mathematica etera, 3 () (13) Y P Thorai, et al, Orderig Geeralized Trapezoidal Fuzzy Numbers, It J Cotemp Math Scieces, 7(1) (1) T Veeraraja, Probability, statistics ad radom process, Tata McGraw ill Educatio Pvt td, New Delhi, Idia 8 R Viertl, Statistical methods for fuzzy data, Joh Wiley ad Sos, Chichester, 11 9 R Viertl, ivariate statistical aalysis with fuzzy data, Computatioal Statistics ad Data alysis, 51 (6) Y M Wag et al, O the cetroids of fuzzy umbers, Fuzzy Sets ad Systems, 157 (6) C Wu, alysis of variace for fuzzy data, Iteratioal Joural of Systems Sciece, 38 (7) C Wu, Statistical cofidece itervals for fuzzy data, Expert Systems with pplicatios, 36 (9) C Wu, Statistical hypotheses testig for fuzzy data, Iformatio Scieces, 175 (5) Zadeh, Fuzzy sets, Iformatio ad Cotrol, 8 (1965)