Modern pplied cience; Vol 8, No 1; 014 IN 1913-1844 E-IN 1913-185 Publihed by Canadian Center of cience and Education Tet of tatitical Hypothee with Repect to a uzzy et P Pandian 1 & D Kalpanapriya 1 1 Department of Mathematic, chool of dvanced cience, VIT Univerity, Vellore-63 014, India Correpondence: P Pandian, Department of Mathematic, chool of dvanced cience, VIT Univerity, Vellore -63014, India E-mail: pandian61@rediffmailcom Received: October 31, 013 ccepted: November 7, 013 Online Publihed: December 17, 013 doi:105539/mav8n1p35 URL: http://dxdoiorg/105539/mav8n1p35 btract Tet of tatitical hypothee with crip data uing mall ample are extended to with memberhip function of the fuzzy et The t-tet tatitic and the -tet tatitic with repect to fuzzy et are defined uing the memberhip grade of the fuzzy et The rule for taking deciion about the hypothee are provided In the propoed tet, the optimitic and peimitic approach, h-level et, cut and fuzzy interval are not ued Numerical example are provided for undertanding the propoed teting procedure The propoed tet of hypothee may be ueful to deciion maker who are handling real life problem involving linguitic variable / fuzzy et for taking uitable deciion in an acceptable manner Keyword: teting hypothei, mall ample, t-tet, -tet, fuzzy et, memberhip function, confidence limit 1 Introduction tatitical hypothei teting i an applied tatitical analyi in which inference of population parameter are obtained uing the numerical ample of the population The data analyt have intereted to learn tet of tatitical hypothee for analyzing the population parameter In conventional hypothee teting (Devore (008)), conidering ample are crip and the ignificance tet lead to the binary deciion In real life ituation, the ample data can not be recorded preciely alway o, imprecie data ample may be got for teting hypothee Many reearcher (rnold, 1998; Caal et al, 1986; on et al, 199; aade & chwarzlander, 1990; aade, 1994; Caal & Gil, 1989, 1994) have propoed variou tet of tatitical hypothee with imprecie ample Uing the fuzzy data ample, the fuzzy tet of hypothee were dicued in Grzegorzewki (000) and Watanabe and Imaizumi (1993) Nikanen (001) tudied tatitical hypothee in field of human cience Wu (005) developed hypothee tet for fuzzy data uing optimitic and peimitic approach kbari and Rezaei (009) propoed an approach to tet the hypothei about the variance uing fuzzy data or fuzzy data, hypothee tet were tudied by Viertl (006, 011) baed on confidence interval uzzy confidence interval for unknown fuzzy parameter were contructed by Wu (009) or vague data, refi and Taheri (011) teted the tatitical fuzzy hypothee aed on confidence limit, the tatitical hypothee tet for fuzzy data i dicued by Chachi et al (01) The tet of tatitical hypothei for comparing mean with vague data wa conidered aloui Jamkhaneh and Nadi Ghara (010) Kalpanapriya and Pandian (01, 01) propoed tet of hypothei for mean of population uing imprecie ample In thi paper, we propoe four type of tatitical hypothei tet uing mall ample (or ample) baed on the memberhip function (M) of a fuzzy et (or fuzzy et) namely, (i) teting of ignificance for difference of mean of two population with repect to a fuzzy et, (ii) teting of ignificance for difference of mean of a population with repect to two fuzzy et, (iii) to tet the difference of variance of two population with repect to a fuzzy et and (iv) to tet the difference of variance of a population with repect to two fuzzy et The t-tet tatitic and -tet tatitic are defined on the memberhip grade (MG) of the fuzzy et over a random ample The rule for taking deciion about the hypothee are provided The optimitic and peimitic approach, h-level et, cut and fuzzy interval are not ued in the propoed tet The procedure of the propoed tet of hypothee are illutrated by mean of numerical example The propoed tet of hypothee can help deciion maker in the linguitic hypothee tet related iue of real life problem by aiding them in the deciion making proce and providing an appropriate deciion rule in an acceptable manner 5
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 Preliminarie The following concept related to fuzzy et and it M are ued which can be found in George J Klir and o Yuan (008) and Chiang and Lin (1999) Let X and Y be two crip et and let and be fuzzy et where i a fuzzy pace If a fuzzy et i defined on X with a M x ), then can be repreented a follow: ( {( x, ) / x X }, where : X [0,1] If the two fuzzy et and in defined on X with M x ) ( and, then, they are written a follow: {( x, ) / x X } and {( x, ) / x X }, where and : X [0, 1] If a fuzzy et i defined on X and Y with M x ) ( and y ( ), then i written a follow: {( x, ) / x X } and {( y, ( y )) / y Y }, where : X [, ] 01 and : Y [, ] 01 In Zadeh (1968), the probability of a fuzzy et defined on X with M x ) i given by P( ) dp E( ) (1) x where P i the probability meaure over X rom (1), we can conclude that the probability of the happening of the fuzzy event i the expectation of x ) ( if the probability meaure of X i known Now, we need the following definition of the ample mean and the ample variance of the MG of a fuzzy et which can be found in Chiang and Lin (1999) Definition 1: Let { x1,x,,xn } be a random ample of ize n from a crip et X with the MG of a fuzzy et where {( x, ) / x X } Then, the average MG of fuzzy et over the random ample or the ample mean of the M of the fuzzy denoted by i defined a follow: n 1 i n i1 Definition : Let { x1,x,,xn } be a random ample of ize n from a crip et X with the MG of a fuzzy et where {( x, ) / x X } Then, the variance of the MG of fuzzy et over the random ample or the ample variance of the M of the fuzzy denoted by i defined a follow: n 1 i ( ) n 1 i1 and the tandard deviation of the MG of fuzzy et over the random ample or the ample tandard deviation of the M of the fuzzy, 3 Teting of ignificance for Difference of two Population Mean with Repect to uzzy et In thi ection, we propoe the following two type of tet of tatitical hypothee: (i) Tet for the difference of mean of two population uing their mall ample with repect to a fuzzy et (ii) Tet for the difference of mean of a population uing it mall ample with repect to two fuzzy et 31 Teting of ignificance for the Difference of two Population Mean with Repect to a uzzy et Let X and Y be two crip population and be a fuzzy et defined on X and Y Let { x 1, x,, x m } be a linguitic random ample of X with MG x ) ( i, i=1,,,m and { y 1, y,, y n } be another linguitic random ample of Y with MG y ) ( j, j=1,,,n uch that m n 30 aed on the ample, we tet that the mean of the population X with repect to, (, X ) and the mean of the population Y with repect to, (,Y ) are the ame Let the ample mean of the M of defined on X be, the ample mean of the M of defined on Y be ( y ), the ample variance of the M of defined on X be and the ample variance of the M of defined on Y be ( y ) Now, we have the null hypothei (NH), H : (, X ) (,Y ) If both population have ame tandard deviation with repect to, we ue the tet tatitic for teting the NH, 6 (
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 t ( y ) 1 1 m n ( m 1) ( n 1) ( y ) where m n If both population tandard deviation with repect to are not the ame, we ue the tet tatitic for teting the NH, t ( y ) ( y ) m n Now, the degree of freedom (df) ued in thi tet i n m Let the level of ignificance (LO) be and let t, denote the table value of t for df at level Now, for LO, the critical region of the alternative hypothei (H), H i given below: lternative Hypothei Critical Region (, X ) (,Y ) (upper tailed tet) t t, v (, X ) (,Y ) (lower tailed tet) t t, v (, X ) (,Y ) (two tailed tet) t t /, v If t t, v (one tailed tet), the difference between (, X ) and (,Y ) at level i not ignificant That i, (, X ) (,Y ) (the mean of population with repect to are identical) at level Therefore, the NH i accepted Otherwie, the H ( (, X ) (,Y ) (for upper tailed tet) or (, X ) (,Y ) (for lower tailed tet)) i accepted If t t /, v (two tailed tet), the difference between (, X ) and (,Y ) at level i not ignificant That i, (, X ) (,Y ) (the population mean of the M of are identical) at level Therefore, the NH i accepted Otherwie, the H, that i, (, X ) (,Y ) i accepted Now, the 100(1 )% confidence limit for the difference of the population mean of the M of, (, X ) and (,Y ) correponding to the given ample are given below: 1 1 1 1 ( ( y )) tt (,X) - (,Y ) ( T (x) ( y)) t m n m n ( y ) ( y ) ( ( y )) t (, X) - (, Y) ( ( y )) t T T m n m n where t T t /, v Now, with the help of the numerical example given below, the procedure of the above aid teting of hypothei i explained Example 31: Let X = {tudent in a Government Univerity } and Y = {tudent in a Private Univerity } be the two population Let the fuzzy et, = { comfort ability} be defined on X and Y Now, we are going to tet that the in X i better than the in Y, that i, (, X ) (, Y ) Let 1 { x1,x,x3,x4,x5,x6,x7,x8 } be the ample of ize eight taken from the population X and { y1, y, y3, y4, y5, y6, y7 } be the ample of ize even taken from the population Y 7
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 Then, the memberhip grade of the given two ample baed on their information concerning the fuzzy et are given below x i x 1 x x3 x4 x5 x6 x7 x 8 x ) ( i 091 085 08 079 089 076 081 078 and y j y 1 y y 3 y 4 y 5 y 6 y 7 y ) ( j 074 076 06 079 086 065 058 Now, the ample average memberhip grade of and the ample variance of over thee two ample are 086 ; ( y ) 0 714 ; 0 0087 and ( y ) 0 01019 n n ( y ) Now, 1 8 7 0 085 Now the NH, Ho : ( X ) (Y ) againt the H, H : ( X ) (Y ) We take LO, 5% and the table value of t for 13 df at 5% LO (one tailed tet) i 16 Now, the tet tatitic, ( y ) t 086 0714 = 545 16 1 1 1 1 0085 n1 n 8 7 Therefore, the NH i rejected and the H i accepted Thu, on the bai of the ample, at 5% LO, the government univerity tudent are more comfortable than private univerity tudent 3 Teting of ignificance for the Difference of Mean of a Population with Repect to two uzzy et Let X be a crip population and and be two fuzzy et defined on X Let { x 1, x,, xm } be a linguitic random ample of ize m from a normal population with MG x ) ( i and ( x i ), i=1,,,m over and repectively aed on the ample, we tet that the mean of the population X related to MG of, (, X ) and the mean of the population X related to MG of, (, X ) are the ame Let the ample mean of the M of and defined on X be and repectively and the ample variance of the M of and defined on X be and Now, we tet the NH, H X ) ( : (,, X ) If population tandard deviation with repect to two fuzzy et are the ame, we ue the tet tatitic for teting the NH, where ( m 1)( ) ( x ) m t m If population tandard deviation with repect to two fuzzy et are not the ame, we ue the tet tatitic for teting the NH, t m Now, the df ued in thi tet i m Let the LO be and let t, be the table value of t for df at level 8
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 Now, for LO, the critical region of the H, H for different type of tet are given below: lternative Hypothei Critical Region (, X ) (, X ) (upper tailed tet) t t, v (, X ) (, X ) (lower tailed tet) t t, v (, X ) (, X ) (two tailed tet) t t /, v If t t, v (one tailed tet), the difference between (, X ) and (, X ) at level i not ignificant That i, the mean of the population X with repect to and the mean of the population X with repect to are the ame ( (, X ) (, X ) ) at level Therefore, the NH i accepted Otherwie, the H ( (, X ) (, X ) (for upper tailed tet) or (, X ) (, X ) (for lower tailed tet)) i accepted If t t /, v (two tailed tet), the difference between (, X ) and (, X ) at level i not ignificant That i, the mean of the population X with repect to and are identical ( (, X ) (, X ) ) at level Therefore, the NH i accepted Otherwie, the H, that i, (, X ) (, X ) i accepted Now, the 100(1 )% confidence limit for the difference of the population mean (, X ) and (, X ) correponding to the given ample are given below: ( ) td (,X) - (,X ) ( m ) td m ( x) ( x) ( x) ( x) ( ( x) ( x)) t (,X) - (,X) ( ( x) ( x)) t D D m m where t D t /,m Now, the numerical example given below i ued to illutrate the above aid hypothei tet procedure Example 3: Let X={Doctor in a city} be the population Let the fuzzy et, = {Compaionate} and ={Contentment} be defined on X We are going to tet that in the city, compaionate doctor and contentment doctor are the ame, that i, ( X ) ( X ) Let { x1,x,x3,x4,x5,x6,x7,x8,x9,x10 } be the ample of ize ten taken randomly The MG of the given ample concerning fuzzy et and baed on the ample information are obtained a Doctor x 1 x x 3 x 4 x 5 x 6 x 7 x 8 x9 x10 x ) ( i 091 085 08 079 089 076 081 078 086 100 ( x i ) 074 076 06 079 07 08 058 07 067 045 Now, the ample average MG of and and the ample variance of the fuzzy et and over the ample are 0 847 ; 0 687 ; 0 00501 and 0 01334 n1 n Now, 0 0987 10 10 Now, the NH, Ho : ( X ) ( X ) and the H, H : ( X ) ( X ) Now, we take LO, = 5% and the table value of t for 18 df at 5% LO (two tailed tet) i 101 Now, the tet tatitic, t 0847 0687 = 3636 101 1 1 1 1 00987 n1 n1 10 10 Therefore, NH i rejected and the H i accepted, that i, the compaionate doctor need not be contentment in the city at 5% LO 9
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 4 Teting of ignificance for Difference of two Variance with Repect to uzzy et The following two type of tet of hypothee are dicued in thi ection: (i) To tet the difference of variance of two population uing their mall ample with repect to a fuzzy et (ii) To tet the difference of variance of one population uing it mall ample with repect to two fuzzy et 41 To Tet the Difference of Variance of two Population with Repect to one uzzy et Let X and Y be two population with variance 1 and repectively Let be a fuzzy et defined on X and Y Let 1 { x 1, x,, xm } be a random ample of ize m from the population X with M x ) ( and { y1, y,, yn} be a random ample of ize n from the population Y with M y ) ( uch that m n 30 Let and ( y ) be ample variance of X and Y with repect to Now, the tet tatitic ( y ) ( y ) ( X ) (Y ) (Y ) ( X ) Now, we have to tet the hypothei that the variance of X and the variance of Y are the ame with repect to, that i, the NH, H : o ( X ) (Y ) Now, the tet tatitic for H o, (v,v )df ( y ) ( y ) with 1 if ( y ) (v,v )df ( y ) with 1 if where v 1 m 1 and v n 1 Let the LO be Now, the critical region of the H, H for LO i given below: Rejection region lternative hypothei ( X ) (Y ) (one tailed tet) (, m,n ) 1 1 ( y ) ( X ) (Y ) (one tailed tet) ( y ),( n1,m 1) ( y ) (two tailed tet) /,( m,n ) 1 1 ( y ) ( y ) /,( n,m ) 1 1 or If ( x) (, m1,n1 ) (one tailed tet), the difference between the variance of X and Y with repect to at ( y) level i not ignificant That i, the population variance with repect to are identical ( ( X ) (Y ) ) at level Therefore, the NH i accepted Otherwie, the H, that i, ( X ) (Y ), i accepted 30
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 If ( y ),( n,m ) 1 1 (one tailed tet), the difference between the variance of X and Y with repect to at level i not ignificant That i, the population variance with repect to are identical ( ( X ) (Y ) ) at level Therefore, the NH i accepted Otherwie, the H, that i, ( X ) (Y ), i accepted If ( y ) /,( n,m ) 1 1 /,( m,n ) 1 1 (two tailed tet), the difference between the variance ( y ) of the population X and Y with repect to at level i not ignificant That i, the variance of the population X with repect to and the variance of the population Y with repect to are the ame ( ( X ) (Y ) ) at level Therefore, the NH i accepted Otherwie, the H, that i, ( X ) (Y ), i accepted Now, the 100(1 )% confidence limit for the quotient of variance ( X ) (Y ) correponding to the given ample are given below: ( y ) ( X ) /,( n,m ) /,( m 1,n 1) (Y ) ( y ) 1 1 ( y ) (Y ) ( y ) /,( n,m ) /,( n 1,m1 ) ( X ) 1 1 Now, we explain the procedure of the above aid hypothei tet with a numerical example Example 41: Let X {ll girl in a city} and Y {ll girl in a town} Now, we tet that the variability of the prettine among girl in both place are the ame Now, a ample of ix girl wa taken at random from the city ( x 1, x, x3, x4, x5, x6) = (Mary, Judy, Linda, uan, etty, Julia) and a ample of ix girl wa taken at random from the town ( y 1, y, y3, y4, y5, y6) = (Maya, Jamine, Latha, hela, indu, Jaya) Now, let u define a fuzzy et over the crip et X and the crip et Y, ={ Pretty girl} Now, the memberhip grade of thee two et of ix girl concerning the fuzzy et baed on the known information are obtained a x Mary Judy Linda uan etty Julia 079 096 065 084 100 088 and y Maya Jamine Latha hela indu Jaya y ) ( 070 098 060 090 083 086 Now, the ample average of are 0 85 and ( y ) 0 81 Now, the ample variance of are given 0 0158 and ( y ) 0 019 Now, the NH, ) ( ) againt the H, ) ( ) X ( Y Now, the tet tatitic: ( y ) X ( Y 31
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 We take 5% and the table value of for v=(5,5) at 5% level (two tailed tet) = 505 ( y ) Now, 0 019 1 5 05 0 0158 Therefore, the NH i accepted at 5% level Thu, city girl and town girl have the ame degree of variation of the prettine, according to the random ample of girl 4 To Tet Variance of one Population with Repect to two uzzy et Let X be a population with variance Let two fuzzy et and be defined on X Let { x, x,, xm be a random ample of the population X of the ize m 30 with M x ) ( and Let and be ample variance of X with repect to and Now, the tet tatitic ( X ) ( X ) ( X ) ( X ) 1 } Now, we have to tet the hypothei that the variance of the population X with repect to and are the ame, that i, the NH, H : ( X ) ( X ) Then, the tet tatitic for H o, o where v 1 m 1 and v m 1 Let the LO be Now, for LO, the critical region of the H, lternative hypothei ( v,v ) df with 1 if ( v,v ) df with 1 if H of variou type of tet are given below: Critical region ( X ) ( X ) (one tailed tet),( m,m ) 1 1 ( X ) ( X ) (one tailed tet),( m1,m 1 ) ( X ) (Y ) (two tailed tet) /,( m,m ) 1 1 /,( m,m ) 1 1 or If,( m1,m 1 ) (one tailed tet), the difference between the variance of X with repect to and at level i not ignificant That, the population variance with repect to and are identical 3
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 ( ( X ) ( X ) ) at level Therefore, the NH i accepted Otherwie, the H, that i, ( X ) ( X ), i accepted If,( m,m ) 1 1 (one tailed tet), the difference between the variance X with repect to and at level i not ignificant That i, the population variance with repect to and are identical ( ( X ) ( X ) ) at level Therefore, the NH i accepted Otherwie, the H, that i, ( X ) ( X ), i accepted If /,( m,m ) 1 1 or /,( m,m ) 1 1 (two tailed tet), the difference between the variance of the population X with repect to and at level i not ignificant That i, the variance of the population X with repect to and the variance of the population X with repect to are the ame ( ( X ) ( X ) ) at level Therefore, the NH i accepted Otherwie, the H, that i, ( X ) ( X ), i accepted Now, the 100(1 )% confidence limit for the quotient of variance ( X ) and ( X ) correponding to the given ample are given below: ( X ) /,( m,m ) /,( m 1,m1 ) ( X ) 1 1 ( X ) /,( m,m ) /,( m 1,m1 ) ( X ) 1 1 Now, we illutrate the procedure of the above aid hypothei tet with help of a numerical example Example 4: Let X = {acultie in a Univerity} be the population Let ( x1, x, x3, x4, x5, x6, x7, x8) be the random ample of X Let the fuzzy et = {Punctuality} and = {vailability} be defined on X Now, the MG of the ample concerning fuzzy et and baed on the collected information are given below aculty x 1 x x 3 x 4 x 5 x 6 x 7 x 8 x ) ( i 096 085 08 079 089 076 081 088 ( x i ) 074 076 06 079 064 08 058 07 We tet the hypothei that facultie in the Univerity have no variability related to the behaviour of punctuality and availability Now, the NH, H : ( X ) ( X ) 0 againt the H, H : ( X ) ( X ) Now, we take 5% and for (7,7) df at 5% level of ignificance, the table value of i 378 Now, the ample average MG of and and the ample variance of and over the ample are 0845 ; 0 709, 0 004086 and 0 007413 Now, the tet tatitic, 0 007413 = 1814 378 0 004086 Therefore, we accept the NH, that i, facultie in the Univerity have no variability related to the behaviour of punctuality and availability 33
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 5 Concluion our type of tet of tatitical hypothee baed on the M of fuzzy et which are totally different from conventional tatitical hypothei teting are propoed in thi article In the propoed tet of hypothee, the difference of mean and variance of the population are tudied with the help of fuzzy et and mall ample of the population The rule for deciion taken about the hypothee are provided We can eaily oberve that the each propoed tet of tatitical hypothei i a characteritic or attribute baed tet on the population The optimitic and peimitic approach, h-level et, cut and fuzzy interval are not ued in the propoed hypothee tet The propoed tatitical hypothee tet can help deciion maker in tet of hypothee related iue of real life problem for chooing an appropriate deciion with atifaction cknowledgement The author thank to Dr M M ahul Hameed, Profeor in Englih, VIT Univerity, Vellore, India for checking the language and the preentation in thi article Reference kbari, M G, & Rezaei, (009) oottrap tatitical inference for the variance baed on fuzzy data utrian Journal of tatitic, 38, 11-130 refi, M, & Taheri, M (011) Teting fuzzy hypothee uing fuzzy data baed on fuzzy tet tatitic Journal of Uncertain ytem, 5, 45-61 http://dxdoiorg/101007/00500-008-0339-3 rnold, (1998) Teting fuzzy hypothee with crip data uzzy et and ytem, 94, 33-333 http://dxdoiorg/pii 0165-01 14(96)0058-8 aloui, J E, & Nadi, G (010) Teting tatitical hypothee for compare mean with vague data International Mathematical orum, 5, 615-60 Caal, M R, Gil, M, & Gil, P (1986) The uzzy deciion problem: an approach to the problem of teting tatitical hypothee with fuzzy information European Journal of Operational Reearch, 7, 371-38 http://dxdoiorg/101016/0377-17(86)90333-4 Caal, M R, & Gil, M (1989) note on the operativene of Neyman Pearon tet with fuzzy information uzzy et and ytem, 30, 15-0 http://dxdoiorg/101016/0165-0114(89)9008-1 Caal, M R, & Gil, P (1994) ayeian equential tet for fuzzy parametric hypothee from fuzzy information Information cience, 80, 83-98 http://dxdoiorg/101016/000-055(94)90080-9 Chachi, J, Taheri, M, & Viertl, R (01) Teting tatitical hypothee baed on fuzzy confidence interval orchungbericht M-01-, Techniche Univeritat Wien, utria Chiang, D, & Lin, N P (1999) Correlation of fuzzy et uzzy et and ytem,, 1-6 Devore, J L (008) Probability and tatitic for Engineer Cengage George, J K, & o Yuan (008) uzzy et and uzzy Logic, Theory and pplication Prentice-Hall, New Jerey Grzegorzewki, P (000) Teting tatitical hypothee with vague data uzzy et and ytem, 11, 501-510 http://dxdoiorg/101016/0165-0114(98)00061-x Kalpanapriya, D, & Pandian, P (01) Two-ample tatitical hypothei tet for mean with imprecie data International Journal of Engineering Reearch and pplication,, 310-317 Kalpanapriya, D, & Pandian, P (01) tatitical hypothee teting with imprecie data pplied Mathematical cience, 6, 585-59 Nikanen, V (001) Propect for oft tatitical computing: decribing data and inferring from data with word in the human cience Information cience, 13, 83-131 http://dxdoiorg/101016/000-055(01)00060-3 aade, J J, & chwarzlander, H (1990) uzzy hypothei teting with hybrid data uzzy et and ytem, 35, 197-1 http://dxdoiorg/101016/0165-0114(90)90193- aade, J J (1994) Extenion of fuzzy hypothei teting with hybrid data uzzy et and ytem, 63, 57-71 http://dxdoiorg/101016/0165-0114(94)90145-7 on, J Ch, ong, I, & Kim, HY (199) fuzzy deciion problem baed on the generalized Neymen-Pearon criterion uzzy et and ytem, 47, 65-75 34
wwwccenetorg/ma Modern pplied cience Vol 8, No 1; 014 Viertl, R (006) Univariate tatitical analyi with fuzzy data Computational tatitic and Data nalyi, 51, 33-147 http://dxdoiorg/101016/jcda0060400 Viertl, R (011) tatitical method for fuzzy data John Wiley and on, Chicheter http://dxdoiorg/10100/9780470974414ch9 Watanabe, N, & Imaizumi, T (1993) fuzzy tatitical tet of fuzzy hypothee uzzy et and ytem, 53, 167-178 Wu, H C (005) tatitical hypothee teting for fuzzy data Information cience, 175, 30-56 http://dxdoiorg/ 101016/jin0031009 Wu, H C (009) tatitical confidence interval for fuzzy data Expert ytem with pplication, 36, 670-67 http://dxdoiorg/ 101016/jewa008010 Zadeh, L (1968) Probability meaure of fuzzy event J Math nal ppl, 3, 41-47 Copyright Copyright for thi article i retained by the author(), with firt publication right granted to the journal Thi i an open-acce article ditributed under the term and condition of the Creative Common ttribution licene (http://creativecommonorg/licene/by/30/) 35