# An Introduction to Fuzzy Testing of Multialternative Hypotheses for Group of Samples with the Single Parameter: Through the Fuzzy Confidence Interval of Region of Acceptance

Authors: Manikandan Harikrishnan; Jeyabharathi Sundarrajan; Muthuraj Rengasamy
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

Classical statistics and many data mining methods rely on “statistical significance” as a sole criterion for evaluating alternative hypotheses. It is very useful to find out the significant difference existing between the samples as well as the population or between two samples. But in this paper,  the researchers try to apply the concepts of fuzzy group testing of hypothesis problem between multi group of samples of same size or different, through comparing the parameters like mean, standard deviation, and so forth. Hence we can compare multigroups such that they have the significant difference in their mean or standard deviation or other parameters through the fuzzy group testing of multihypotheses. The authors introduced and investigated the concepts very first time through fuzzy analysis that can decide which group(s) or samples can be taken for further investigation and eitherH0 is rejected or accepted and hence the next discussion provides the properties of group of samples which may result in the optimized solution for the problem.


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## Body

## 1. Introduction

The fuzzy technique can be used to generate solutions to the problems based on “vague, ambiguous, qualitative, incomplete, or imprecise information.” Fuzzy logic is an extension of fuzzy set theory that has developed since five decades approximately. The problem of dealing with imprecision and uncertainty is a part of the wider human experiences before the introduction of fuzzy theory.In the traditional approach for hypotheses testing, all of the concepts are assumed to be precise and well-defined. But sometimes we have to take decision in an unrealistic manner, because in realistic problems we may come across fuzzy data and fuzzy hypotheses. Sometimes the decisions on samples will not help us to proceed. Arnold [1] studied on fuzzy hypotheses testing with crisp data. The problem of testing fuzzy hypotheses when the observations were crisp was considered by Taheri and Behboodian [2]. Torabi et al. [3] used Neyman-Pearson Lemma for fuzzy hypotheses by testing with vague data. Chachi et al. [4] also studied on the same problem in the context of fuzzy decision problems. Viertl [5] studied testing fuzzy hypotheses using fuzzy data based on fuzzy test statistic.We know that the analysis of variance (ANOVA) is a powerful statistical tool for test of significance between two or more samples. It is used in a situation where three or more samples have been considered at a time, towards the testing of hypothesis that all the samples are drawn from the same population; otherwise their parameters have the significant difference. The purpose of this analysis is to test the homogeneity of several means. This analysis found to test the null hypothesisH0 is that all the means are equal against the alternative hypotheses that some of them are not equal. So the variance ratio
(1)F=variancesbetweenthesamplesvarianceswithinthesamples
which is used for the comparison in the following designs:(1)
completely randomized design (CRD, ANOVA for one-way classification);(2)
randomized block design (RBD, ANOVA for two-way classification);(3)
latin square design (LSD, ANOVA for three-way classification).But these designs suffer by the disadvantage of being inherently less informative than other more sophisticated layouts. This makes the design less efficient and results in less sensitivity in detecting significant effects. For example, if null hypothesis is rejected then it concludes that the treatments differ significantly but it is not clear that which samples could be considered for the further investigation also which are inconsiderable in the future analysis. So we take this for our consideration, and we assign a method which would be very useful to test the null hypothesisH0:θ=θ1=θ2=⋯=θn versus multialternative hypotheses:
(2)AlternativehypothesisH1:θ=θ1=θ2⋯=θn-1≠θnH2:θ=θ1=θ2⋯≠θn-1≠θn··Hϕ:θ≠θ1≠θ2⋯≠θn-1≠θn.In the classical theory of statistical inference there is a one-to-one relationship between the parameter values for which the null hypothesis is accepted and the structure of the confidence intervals. Namely, a family ofα-level acceptance regions for a statistical test concerning the parameter θ is equivalent to a certain 1-α family of confidence intervals for θ. The results of a statistical test, therefore, can alternatively be stated in terms of the corresponding confidence interval. By developing the concepts of fuzzy confidence intervals and fuzzy statistical tests and using the one-to-one relationship between tests and confidence intervals may be convenient to generalize a correspondence between fuzzy confidence interval and fuzzy statistical test. In such a case, we have, for example, the degree of membership of the null hypothesized fuzzy parameter in the fuzzy confidence interval as the degree of acceptability of the null hypothesis. In the following, we use such a relationship in fuzzy environment to test the hypothesis corresponding to the parameters of the multisamples simultaneously via fuzzy group testing of hypothesis by fixing the level of confidence values for different fuzzy hypotheses. In this test two or more numbers of samples can be taken for our consideration and simultaneously we tried here to check the hypothesis whether all the samples coincide with the population parameter or some of the samples have much deviated from the population.


## 2. Fuzzy Group Testing of Hypothesis with Confidence Intervals

### 2.1. The Fuzzy Group Environment

Definition 1 (see [6]).
LetX be any nonempty set. A fuzzy subset μ of X is a function μ:X→[0,1].Definition 2 (see [7]).
A fuzzy setμ on G is called fuzzy subgroup of G if for x,y∈G,(i)
μ(xy)≥min⁡{μ(x),μ(y)},(ii)
μ(x-1)=μ(x).Definition 3 (see [8]).
LetG be a finite group. In 2G-{ϕ}, a nonempty set ϑ⊂2G-{ϕ} is called a HX group on G, if ϑ is a group with respect to the algebraic operation defined by AB={ab/a∈A and b∈B}, in which its unit element is denoted by E.Definition 4 (see [8]).
Letμ be a fuzzy subset defined on G. Let ϑ⊂2G-{ϕ} be a HX group on G. A fuzzy set λμ defined on ϑ is said to be a fuzzy subgroup induced by μ on ϑ or a fuzzy HX subgroup on ϑ, if, for any A,B∈ϑ,(i)
λμ(AB)≥min⁡{λμ(A),λμ(B)},(ii)
λμ(A-1)=λμ(A),where λμ(A)=max⁡{μ(x)/forallx∈A⊆G}.Assumptions. Let (Ω,A(ϑi),₱) be a probability space (where Ω is a set of all possible outcomes of an experiment), ϑi⊆2G-ϕ (collection of subsets of G such that G1,G2,…⊆ϑi or ϑi is collection of Gi), and ₱ is a set of probability measure. A(ϑi) is the collection of λi where λi:Gi→F(R) is a fuzzy valued function and Gi is a random sample having distribution fθi_ with parameter θi_=(θi1,θi2,θi3,…,θin), for all i=1,2,…,k.The fuzzy valued functionλi:Gi→F(R) is a fuzzy random variable if Xi(h)l:Gi→F(R) and Xi(h)u:Gi→F(R) are two real valued random variables for all h∈[0,1] where ∀x∈Gi;λih(x)=[Xihl,Xihu].Fuzzy random variablesλ1 and λ2 are identical if Xihl,Yihl and Xihu,Yihu are identical for all h∈[0,1]. If θ=θij for all i=1,2,… for some j, then θij is known as identity element of the group θi. If all θi are elements of the HX group ϑ, then the comparison may vary between the different collection of θi and the identity element of ϑ be E=θk which consisting the identity element “e” of G. Also it is very clear that λij(e)≥λij(x) for all i. The element “e” is called reaching height element (RHT) of each group of samples.Definition 5.
Here let us considerG1, and G2 be the two samples contain equal number of elements, considered for group testing of hypothesis. We say that it is a fuzzy random sample of size “n” from fθi_, if λij’s are independent and identically distributed fuzzy random variables from fθi_ where j=1,2,…n. It is very clear that from the definition of independent and identically distributed (i.i.d) fuzzy random variable (FRV), for fuzzy random sample, λi1,λi2,λi3,…,λin→i.i.dfθi_ for i=1,2, we can see that λi1l,λi2l,λi3l,…,λinl→i.i.dfθi_l and λi1u,λi2u,λi3u,…,λinu→i.i.dfθi_u for all h∈[0,1].


### 2.2. Group Testing of Hypotheses: Classical Approach

Consider the problem of testing the null hypothesisH0:θ=θ1=θ2 versus multialternative hypotheses:
(3)AlternativehypothesisH1:θ=θ1≠θ2,H2:θ=θ2≠θ1,Hϕ:θ≠θ1≠θ2.
For each θ=θ1=θ2, let A (θ=θ1=θ2) denote the acceptance region of a level α-group testing of hypothesis H0:θ=θ1=θ2 and α=αK, where K=Ω or 1 or 2 or j.IfSi(X)={θi:Xi∈A(θi)} then
(4)θi∈Si(X)⟺Xi∈A(θi),
and hence
(5)Pθiθi∈SiX≥1-α,∀θi.
We can define the area of level of significance for different hypotheses as follows:
(6)Ari=ArΩ=AS1Xθi∩AS2Xθi,Ar1=AS1Xθi,Ar2=AS2Xθi,Ar3=R-AS1Xθi∪AS2Xθi.
The indicator function ISζ(X)(θ) is defined as
(7)θ∈SζXthenISζXθ=1,θ∉Sζ(X)then1-ISζX(θ)=1.So, a confidence set can be viewed as a statement about testing hypothesisHi, which exhibits the values for which the hypothesis is completely accepted if
(8)φ(X)=ISΩ(X)(X)AcceptH0,IS1(X)(X)AcceptH1,IS2(X)(X)AcceptH2,ISϕ(X)(X)AcceptHϕ.The above said value of the test is a rule stating that the null hypothesis cannot be rejected if the interval contains the hypothesized value and can be rejected if the other intervals completely contain the hypothesized value of the given problem. So here directly we can have the chance for making the conclusion about the multihypotheses using the hypothesized value and the confidence interval of respective hypothesis.Example 6.
LetG1=X11,X12,X13,…,X1n} and G2=X21,X22,X23,…,X2n} be →i.i.d from normal distribution N(θ,1) with the unknown means θ1 and θ2, respectively. A confidence interval at confidence level 1-α=1-αK, where K=Ω or 1 or 2 or ϕ for θi of the form
(9)SKXi=X¯1-1nZ1-α2,X¯1+1nZ1-α2∩X¯2-1nZ2-α2,X¯2+1nZ2-α2X¯2+1nZ1-α2,X¯1+1nZ1-α2X¯2-1nZ2-α2,X¯1-1nZ2-α2R-X¯1-1nZ1-α2,X¯1+1nZ1-α2∪X¯2-1nZ2-α2,X¯2+1nZ2-α2.
This can be derived easily, where Zα is the α-quartile of standard normal distribution.
Assume the two random samples with sizesn=21, X¯1=0.8, and X¯2=0.75 are observed and we want to test H0(0.6):θ=0.6 versus the other alternative hypotheses H1, H2, and Hφ at level α=0.05. Then we can have
(10)SK(Xi)=(0.39729,1.20270)∩(0.34729,1.15270)(1.15270,1.20270)(0.34729,0.39729)-∞,0.34729∪1.20270,∞,
where Zα=1.96 and K=Ω, 1, 2, and ϕ.
Hence the graphical representations ofSK(Xi)’s are given in Figure 1.Figure 1
Graphical representations ofH0 and the alternative hypotheses.Consider the approximate test statistic(11)Zc=2X¯-(X¯1+X¯2)102-14S2/N+(s12+s22/n1+n2),if2samplesaregiven,Zc=3X¯-X¯1+X¯2+X¯3103-19S2/N+s12+s22+s32/n1+n2+n3,if3samplesaregiven,
where X¯ = population mean (parameter), X¯1,X¯2,X¯3 = sample means, S2 = population variance, s12,s22,s32 = variances of samples, and n1,n2,n3 = sample sizes.Hence we have(12)φX=1AcceptH0,0RejectH1,0RejectH2,0RejectHϕ0RejectH0,1AcceptH1,0RejectH2,0RejectHϕ0RejectH0,0RejectH1,1AcceptH2,0RejectHϕ0RejectH0,0RejectH1,0RejectH2,1AcceptHϕ.In this case, based on the observed value ofX¯1 and X¯2, we accept the null hypotheses at level of α=0.05 but the rejection of null hypothesis does not mean H1 as accepted.


### 2.3. The Proposal in Fuzzy Environment

In this section, we investigate a procedure to provide a fuzzy test for group testing of a fuzzy parameter for fuzzy data which is based on fuzzy confidence intervals. In order to derive the degrees of acceptability of the null and alternative hypotheses, we introduce the following method in this test. It is very clear that the data available are observations of a normal fuzzy random sample with the unknown fuzzy meanθ and known variance σ2.That is,X11,X12,…,X1n→i.i.dN(θ,σ2) and X21,X22,…,X2n→i.i.dN(θ,σ2). We restrict our attention only for two sided alternative hypotheses.(1) First we transform the original problem(13)H0θ¯:θ¯=θ¯1=θ¯2versusAlternativehypothesesH1(θ¯):θ¯=θ¯1≠θ¯2H2(θ¯):θ¯=θ¯2≠θ¯1Hϕ(θ¯):θ¯≠θ¯1≠θ¯2
into a set of crisp group testing problems concerning h-levels of the fuzzy parameter. For each h level, based on the samples,
(14)X1hl=X11hl,X12hl,…,X1nhl,X2hl=X21hl,X22hl,…,X2nhl,X1hu=X11hu,X12hu,…,X1nhu,X2hu=X21hu,X22hu,…,X2nhu.
The following classical group testing problems are solved at level α:
(15)H0hl:θ¯hl=θ¯1hl=θ¯2hlversusH1hl:θ¯hl=θ¯1hl≠θ¯2hlH2hl:θ¯hl=θ¯2hl≠θ¯1hlHϕhl:θ¯hl≠θ¯1hl≠θ¯2hlH0hu:θ¯hu=θ¯1hu=θ¯2huversusH1hu:θ¯hu=θ¯1hu≠θ¯2huH2hu:θ¯hu=θ¯2hu≠θ¯1huHϕhu:θ¯hu≠θ¯1hu≠θ¯2hu.We obtain the1-α confidence intervals for the crisp parameters θhl,θhu for each h∈[0,1], denoted by [L1(X1hl),L2(X1hl)], [L1(X2hl),L2(X2hl)] and [U1(X1hl),U2(X1hl)], [U1(X2hl),U2(X2hl)], respectively. The test functions are defined as follows:
(16)φ(X1hl)=I[L1(X1hl),L2(X1hl)](X)AcceptH0,I1-[L1(X1hl),L2(X1hl)](X)RejectH0,φX1hu=IU1X1hu,U2X1huXAcceptH0,I1-U1X1hu,U2X1huXRejectH0⟶1φ(X2hl)=I[L1(X2hl),L2(X2hl)](X)AcceptH0,I1-[L1(X2hl),L2(X2hl)](X)RejectH0φX1hu=IU1X2hu,U2X2huXAcceptH0,I1-U1X2hu,U2X2huXRejectH0⟶2.Let us consider two brands of sizes 21 to be compared with the population such that their average life differs or coincides. Consider the average life time of the population to be around 20,000 working life time with the variance 4,00,000 represented by a triangular fuzzy numberθ¯0=(20000,500,500)T. We consider the physical problem of testing the hypothesis H0:θ0=θ10=θ20 at level α=0.05, using fuzzy group testing of hypotheses. We have
(17)X1=18850,165,180T,X2=19760,154,172T,X1h=[X1hl,X1hu]=[18685+165h,19030-180h],X2h=X2hl,X2hu=19606+154h,19932-172h,θ¯0h=θ0hl,θ0hu=19500+500h,20500-500h.
Hence, the two sided 0.95 confidence intervals for the θ1hl,θ1hu and θ2hl,θ2hu are(i)
θ1hl=[18543.58+165h,18826.42+165h] and θ1hu=[18888.58-180h,19171.42-180h];(ii)
θ2hl=[19464.58+154h,19747.42+154h] and θ2hu=[19790.58-172h,20073.42-172h].Similarly, we can also calculate the membership values of the variablesX1 and X2. Then the membership value of fuzzy confidence interval CT is defined as CT(θ0)l=Wl/Wl+Sl, CT(θ0)u=Wu/Wu+Su.Here the values ofWl, Wu, Sl, and Su are calculated by using the formula given in Appendix A.The degree of membership ofθ0¯=(20000,500,500)T in two sided confidence interval is
(18)CTθ0l=WlWl+Sl,whereWl=W1hl+W2hl,Sl=S1hl+S2hl,CTθ0u=WuWu+Su,whereWu=W1hu+W2hu,Su=S1hu+S2hu.
Therefore we have the confidence interval for accepting H0 as [CT(θ0)l,CT(θ0)u] (Figure 2).Figure 2
Graphical representation of confidence interval forH0 (graph for SΩ).LetCT(θ0)=W/(W+S), where W=W1hl+W1hu+W2hl+W2hu and S=S1hl+S1hu+S2hl+S2hu.(1)
IfCTθ0=0.325 (say) lies inside [CT(θ0)l,CT(θ0)u], then briefly the function of acceptance of H0 is
(19)SΩX=0.325AcceptH0,0.675RejectH0.(2)
Hence we can calculate the membership values ofθ10 and θ20 by using CT(θ10)=(W1)/(W1+S1);CT(θ20)=(W2)/(W2+S2) where W1=W1l+W1u;S1=S1l+S1u and W2=W2l+W2u;S2=S2l+S2u.ConsiderCTθ10=0.156 and CT(θ20)=0.195 and the rejection of H0 may result as
(20)SKX=0.675RejectH0,0.156AcceptH1,0RejectH2,0RejectHϕ0.675RejectH0,0.675-0.156RejectH1,0.195AcceptH2,0RejectHϕ0.675RejectH0,0.675-0.156RejectH1,0.675-0.195RejectH2,0.675-(0.156+0.195)RejectHϕ.(3)
IfCT(θ¯10) lies within the interval of CT(θ¯0) and CT(θ¯20) which lies partially outside of CTθ¯0, then the first group can be accepted and the second can be rejected with the help of fuzzy membership value of CT(θ¯0). Hence H1 is accepted (Figure 3).(4)
IfCT(θ¯10) lies outside the interval of CT(θ¯0) and CT(θ¯20) lies inside CT(θ¯0), then the first sample can be rejected and the second may be accepted. Hence H2 is partially accepted (Figure 4).(5)
IfCT(θ¯10) and CT(θ¯20) both partially lie inside, then it is very clear that both the samples have significant difference by comparing the population parameter. Hence Hφ can be chosen (Figure 5).Figure 3
Graphical representation of confidence interval forH1 (graph for S1).Figure 4
Graphical representation of confidence interval forH2 (graph for S2).Figure 5
Graphical representation for acceptingHφ (rejection of all samples) (graph for Sφ).


## 2.1. The Fuzzy Group Environment

Definition 1 (see [6]).
LetX be any nonempty set. A fuzzy subset μ of X is a function μ:X→[0,1].Definition 2 (see [7]).
A fuzzy setμ on G is called fuzzy subgroup of G if for x,y∈G,(i)
μ(xy)≥min⁡{μ(x),μ(y)},(ii)
μ(x-1)=μ(x).Definition 3 (see [8]).
LetG be a finite group. In 2G-{ϕ}, a nonempty set ϑ⊂2G-{ϕ} is called a HX group on G, if ϑ is a group with respect to the algebraic operation defined by AB={ab/a∈A and b∈B}, in which its unit element is denoted by E.Definition 4 (see [8]).
Letμ be a fuzzy subset defined on G. Let ϑ⊂2G-{ϕ} be a HX group on G. A fuzzy set λμ defined on ϑ is said to be a fuzzy subgroup induced by μ on ϑ or a fuzzy HX subgroup on ϑ, if, for any A,B∈ϑ,(i)
λμ(AB)≥min⁡{λμ(A),λμ(B)},(ii)
λμ(A-1)=λμ(A),where λμ(A)=max⁡{μ(x)/forallx∈A⊆G}.Assumptions. Let (Ω,A(ϑi),₱) be a probability space (where Ω is a set of all possible outcomes of an experiment), ϑi⊆2G-ϕ (collection of subsets of G such that G1,G2,…⊆ϑi or ϑi is collection of Gi), and ₱ is a set of probability measure. A(ϑi) is the collection of λi where λi:Gi→F(R) is a fuzzy valued function and Gi is a random sample having distribution fθi_ with parameter θi_=(θi1,θi2,θi3,…,θin), for all i=1,2,…,k.The fuzzy valued functionλi:Gi→F(R) is a fuzzy random variable if Xi(h)l:Gi→F(R) and Xi(h)u:Gi→F(R) are two real valued random variables for all h∈[0,1] where ∀x∈Gi;λih(x)=[Xihl,Xihu].Fuzzy random variablesλ1 and λ2 are identical if Xihl,Yihl and Xihu,Yihu are identical for all h∈[0,1]. If θ=θij for all i=1,2,… for some j, then θij is known as identity element of the group θi. If all θi are elements of the HX group ϑ, then the comparison may vary between the different collection of θi and the identity element of ϑ be E=θk which consisting the identity element “e” of G. Also it is very clear that λij(e)≥λij(x) for all i. The element “e” is called reaching height element (RHT) of each group of samples.Definition 5.
Here let us considerG1, and G2 be the two samples contain equal number of elements, considered for group testing of hypothesis. We say that it is a fuzzy random sample of size “n” from fθi_, if λij’s are independent and identically distributed fuzzy random variables from fθi_ where j=1,2,…n. It is very clear that from the definition of independent and identically distributed (i.i.d) fuzzy random variable (FRV), for fuzzy random sample, λi1,λi2,λi3,…,λin→i.i.dfθi_ for i=1,2, we can see that λi1l,λi2l,λi3l,…,λinl→i.i.dfθi_l and λi1u,λi2u,λi3u,…,λinu→i.i.dfθi_u for all h∈[0,1].


## 2.2. Group Testing of Hypotheses: Classical Approach

Consider the problem of testing the null hypothesisH0:θ=θ1=θ2 versus multialternative hypotheses:
(3)AlternativehypothesisH1:θ=θ1≠θ2,H2:θ=θ2≠θ1,Hϕ:θ≠θ1≠θ2.
For each θ=θ1=θ2, let A (θ=θ1=θ2) denote the acceptance region of a level α-group testing of hypothesis H0:θ=θ1=θ2 and α=αK, where K=Ω or 1 or 2 or j.IfSi(X)={θi:Xi∈A(θi)} then
(4)θi∈Si(X)⟺Xi∈A(θi),
and hence
(5)Pθiθi∈SiX≥1-α,∀θi.
We can define the area of level of significance for different hypotheses as follows:
(6)Ari=ArΩ=AS1Xθi∩AS2Xθi,Ar1=AS1Xθi,Ar2=AS2Xθi,Ar3=R-AS1Xθi∪AS2Xθi.
The indicator function ISζ(X)(θ) is defined as
(7)θ∈SζXthenISζXθ=1,θ∉Sζ(X)then1-ISζX(θ)=1.So, a confidence set can be viewed as a statement about testing hypothesisHi, which exhibits the values for which the hypothesis is completely accepted if
(8)φ(X)=ISΩ(X)(X)AcceptH0,IS1(X)(X)AcceptH1,IS2(X)(X)AcceptH2,ISϕ(X)(X)AcceptHϕ.The above said value of the test is a rule stating that the null hypothesis cannot be rejected if the interval contains the hypothesized value and can be rejected if the other intervals completely contain the hypothesized value of the given problem. So here directly we can have the chance for making the conclusion about the multihypotheses using the hypothesized value and the confidence interval of respective hypothesis.Example 6.
LetG1=X11,X12,X13,…,X1n} and G2=X21,X22,X23,…,X2n} be →i.i.d from normal distribution N(θ,1) with the unknown means θ1 and θ2, respectively. A confidence interval at confidence level 1-α=1-αK, where K=Ω or 1 or 2 or ϕ for θi of the form
(9)SKXi=X¯1-1nZ1-α2,X¯1+1nZ1-α2∩X¯2-1nZ2-α2,X¯2+1nZ2-α2X¯2+1nZ1-α2,X¯1+1nZ1-α2X¯2-1nZ2-α2,X¯1-1nZ2-α2R-X¯1-1nZ1-α2,X¯1+1nZ1-α2∪X¯2-1nZ2-α2,X¯2+1nZ2-α2.
This can be derived easily, where Zα is the α-quartile of standard normal distribution.
Assume the two random samples with sizesn=21, X¯1=0.8, and X¯2=0.75 are observed and we want to test H0(0.6):θ=0.6 versus the other alternative hypotheses H1, H2, and Hφ at level α=0.05. Then we can have
(10)SK(Xi)=(0.39729,1.20270)∩(0.34729,1.15270)(1.15270,1.20270)(0.34729,0.39729)-∞,0.34729∪1.20270,∞,
where Zα=1.96 and K=Ω, 1, 2, and ϕ.
Hence the graphical representations ofSK(Xi)’s are given in Figure 1.Figure 1
Graphical representations ofH0 and the alternative hypotheses.Consider the approximate test statistic(11)Zc=2X¯-(X¯1+X¯2)102-14S2/N+(s12+s22/n1+n2),if2samplesaregiven,Zc=3X¯-X¯1+X¯2+X¯3103-19S2/N+s12+s22+s32/n1+n2+n3,if3samplesaregiven,
where X¯ = population mean (parameter), X¯1,X¯2,X¯3 = sample means, S2 = population variance, s12,s22,s32 = variances of samples, and n1,n2,n3 = sample sizes.Hence we have(12)φX=1AcceptH0,0RejectH1,0RejectH2,0RejectHϕ0RejectH0,1AcceptH1,0RejectH2,0RejectHϕ0RejectH0,0RejectH1,1AcceptH2,0RejectHϕ0RejectH0,0RejectH1,0RejectH2,1AcceptHϕ.In this case, based on the observed value ofX¯1 and X¯2, we accept the null hypotheses at level of α=0.05 but the rejection of null hypothesis does not mean H1 as accepted.


## 2.3. The Proposal in Fuzzy Environment

In this section, we investigate a procedure to provide a fuzzy test for group testing of a fuzzy parameter for fuzzy data which is based on fuzzy confidence intervals. In order to derive the degrees of acceptability of the null and alternative hypotheses, we introduce the following method in this test. It is very clear that the data available are observations of a normal fuzzy random sample with the unknown fuzzy meanθ and known variance σ2.That is,X11,X12,…,X1n→i.i.dN(θ,σ2) and X21,X22,…,X2n→i.i.dN(θ,σ2). We restrict our attention only for two sided alternative hypotheses.(1) First we transform the original problem(13)H0θ¯:θ¯=θ¯1=θ¯2versusAlternativehypothesesH1(θ¯):θ¯=θ¯1≠θ¯2H2(θ¯):θ¯=θ¯2≠θ¯1Hϕ(θ¯):θ¯≠θ¯1≠θ¯2
into a set of crisp group testing problems concerning h-levels of the fuzzy parameter. For each h level, based on the samples,
(14)X1hl=X11hl,X12hl,…,X1nhl,X2hl=X21hl,X22hl,…,X2nhl,X1hu=X11hu,X12hu,…,X1nhu,X2hu=X21hu,X22hu,…,X2nhu.
The following classical group testing problems are solved at level α:
(15)H0hl:θ¯hl=θ¯1hl=θ¯2hlversusH1hl:θ¯hl=θ¯1hl≠θ¯2hlH2hl:θ¯hl=θ¯2hl≠θ¯1hlHϕhl:θ¯hl≠θ¯1hl≠θ¯2hlH0hu:θ¯hu=θ¯1hu=θ¯2huversusH1hu:θ¯hu=θ¯1hu≠θ¯2huH2hu:θ¯hu=θ¯2hu≠θ¯1huHϕhu:θ¯hu≠θ¯1hu≠θ¯2hu.We obtain the1-α confidence intervals for the crisp parameters θhl,θhu for each h∈[0,1], denoted by [L1(X1hl),L2(X1hl)], [L1(X2hl),L2(X2hl)] and [U1(X1hl),U2(X1hl)], [U1(X2hl),U2(X2hl)], respectively. The test functions are defined as follows:
(16)φ(X1hl)=I[L1(X1hl),L2(X1hl)](X)AcceptH0,I1-[L1(X1hl),L2(X1hl)](X)RejectH0,φX1hu=IU1X1hu,U2X1huXAcceptH0,I1-U1X1hu,U2X1huXRejectH0⟶1φ(X2hl)=I[L1(X2hl),L2(X2hl)](X)AcceptH0,I1-[L1(X2hl),L2(X2hl)](X)RejectH0φX1hu=IU1X2hu,U2X2huXAcceptH0,I1-U1X2hu,U2X2huXRejectH0⟶2.Let us consider two brands of sizes 21 to be compared with the population such that their average life differs or coincides. Consider the average life time of the population to be around 20,000 working life time with the variance 4,00,000 represented by a triangular fuzzy numberθ¯0=(20000,500,500)T. We consider the physical problem of testing the hypothesis H0:θ0=θ10=θ20 at level α=0.05, using fuzzy group testing of hypotheses. We have
(17)X1=18850,165,180T,X2=19760,154,172T,X1h=[X1hl,X1hu]=[18685+165h,19030-180h],X2h=X2hl,X2hu=19606+154h,19932-172h,θ¯0h=θ0hl,θ0hu=19500+500h,20500-500h.
Hence, the two sided 0.95 confidence intervals for the θ1hl,θ1hu and θ2hl,θ2hu are(i)
θ1hl=[18543.58+165h,18826.42+165h] and θ1hu=[18888.58-180h,19171.42-180h];(ii)
θ2hl=[19464.58+154h,19747.42+154h] and θ2hu=[19790.58-172h,20073.42-172h].Similarly, we can also calculate the membership values of the variablesX1 and X2. Then the membership value of fuzzy confidence interval CT is defined as CT(θ0)l=Wl/Wl+Sl, CT(θ0)u=Wu/Wu+Su.Here the values ofWl, Wu, Sl, and Su are calculated by using the formula given in Appendix A.The degree of membership ofθ0¯=(20000,500,500)T in two sided confidence interval is
(18)CTθ0l=WlWl+Sl,whereWl=W1hl+W2hl,Sl=S1hl+S2hl,CTθ0u=WuWu+Su,whereWu=W1hu+W2hu,Su=S1hu+S2hu.
Therefore we have the confidence interval for accepting H0 as [CT(θ0)l,CT(θ0)u] (Figure 2).Figure 2
Graphical representation of confidence interval forH0 (graph for SΩ).LetCT(θ0)=W/(W+S), where W=W1hl+W1hu+W2hl+W2hu and S=S1hl+S1hu+S2hl+S2hu.(1)
IfCTθ0=0.325 (say) lies inside [CT(θ0)l,CT(θ0)u], then briefly the function of acceptance of H0 is
(19)SΩX=0.325AcceptH0,0.675RejectH0.(2)
Hence we can calculate the membership values ofθ10 and θ20 by using CT(θ10)=(W1)/(W1+S1);CT(θ20)=(W2)/(W2+S2) where W1=W1l+W1u;S1=S1l+S1u and W2=W2l+W2u;S2=S2l+S2u.ConsiderCTθ10=0.156 and CT(θ20)=0.195 and the rejection of H0 may result as
(20)SKX=0.675RejectH0,0.156AcceptH1,0RejectH2,0RejectHϕ0.675RejectH0,0.675-0.156RejectH1,0.195AcceptH2,0RejectHϕ0.675RejectH0,0.675-0.156RejectH1,0.675-0.195RejectH2,0.675-(0.156+0.195)RejectHϕ.(3)
IfCT(θ¯10) lies within the interval of CT(θ¯0) and CT(θ¯20) which lies partially outside of CTθ¯0, then the first group can be accepted and the second can be rejected with the help of fuzzy membership value of CT(θ¯0). Hence H1 is accepted (Figure 3).(4)
IfCT(θ¯10) lies outside the interval of CT(θ¯0) and CT(θ¯20) lies inside CT(θ¯0), then the first sample can be rejected and the second may be accepted. Hence H2 is partially accepted (Figure 4).(5)
IfCT(θ¯10) and CT(θ¯20) both partially lie inside, then it is very clear that both the samples have significant difference by comparing the population parameter. Hence Hφ can be chosen (Figure 5).Figure 3
Graphical representation of confidence interval forH1 (graph for S1).Figure 4
Graphical representation of confidence interval forH2 (graph for S2).Figure 5
Graphical representation for acceptingHφ (rejection of all samples) (graph for Sφ).


## 3. Conclusion

This analysis can be extended for multi groups (more than two) and we can check the chosen groups are coinciding with the population parameter or within the groups. So in a single attempt, we can decide the suitable samples for our further investigation according to the necessity, instead of rejection of null hypothesis. This is an important advantage of this method, so we simultaneously can make the decision that would be better to improve the property analyze of the variables in future. Classical approach of a statistical problem will not be very precise sometimes and it will not help us to make better decisions about the properties of the variables. Many of the methods like this were already introduced by mathematicians to solve problems using concepts of fuzzy. So we try to find one of the solutions for the testing of hypotheses problem through fuzzy group testing of hypotheses which may help us in this aspect and also it may be very useful to make the broad decisions upon the groups of samples.So through this discussion, we may get the following conclusions.(1)
This methodology can be extended to compare more than two groups; meanwhile multialternative hypotheses can be discussed and further property analysis of the variables can also be very effective for future discussion.(2)
This can be extended in future for collection of multi-subgroups of a special group called HX group. For example, number of groups (systems) can be developed by choosing different collection of elements of a single group which can be analyzed by using fuzzy group testing method to identify the better performing system(s) to complete the task in the successful manner as well as to consume time. This may be very useful to induce the performance level to the expected height. These are the expected advantages of this research work in future.



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*Source: 365304-2015-02-08.xml*