Tests of Statistical Hypotheses with Respect to a Fuzzy Set

Tests of statistical hypotheses with crisp data using small samples are extended to with membership function of the fuzzy sets. The t-test statistic and the F-test statistic with respect to fuzzy sets are defined using the membership grades of the fuzzy sets. The rules for taking decision about the hypotheses are provided. In the proposed tests, the optimistic and pessimistic approach, h-level set,   cut and fuzzy interval are not used. Numerical examples are provided for understanding the proposed testing procedures. The proposed tests of hypotheses may be useful to decision makers who are handling real life problems involving linguistic variables / fuzzy sets for taking suitable decisions in an acceptable manner.


Introduction
Statistical hypothesis testing is an applied statistical analysis in which inference of populations parameters are obtained using the numerical samples of the populations.The data analysts have interested to learn tests of statistical hypotheses for analyzing the population parameters.In conventional hypotheses testing (Devore (2008)), considering samples are crisp and the significance test leads to the binary decision.In real life situations, the sample data can not be recorded precisely always.So, imprecise data sample may be got for testing hypotheses.Many researchers (Arnold, 1998;Casals et al., 1986;Son et al., 1992;Saade & Schwarzlander, 1990;Saade, 1994;Casals & Gil, 1989, 1994) have proposed various tests of statistical hypotheses with imprecise samples.Using the fuzzy data sample, the fuzzy tests of hypotheses were discussed in Grzegorzewski (2000) and Watanabe and Imaizumi (1993).Niskanen (2001) studied statistical hypotheses in fields of human sciences.Wu (2005) developed hypotheses tests for fuzzy data using optimistic and pessimistic approach.Akbari and Rezaei (2009) proposed an approach to test the hypothesis about the variance using fuzzy data.For fuzzy data, hypotheses tests were studied by Viertl (2006Viertl ( , 2011) ) based on confidence intervals.Fuzzy confidence intervals for unknown fuzzy parameters were constructed by Wu (2009).For vague data, Arefi and Taheri (2011) tested the statistical fuzzy hypotheses.Based on confidence limits, the statistical hypotheses tests for fuzzy data is discussed by Chachi et al. (2012).The test of statistical hypothesis for comparing means with vague data was considered Baloui Jamkhaneh and Nadi Ghara (2010).Kalpanapriya andPandian (2012, 2012) proposed tests of hypothesis for means of populations using imprecise samples.
In this paper, we propose four types of statistical hypothesis tests using small sample (or samples) based on the membership function (MF) of a fuzzy set (or fuzzy sets) namely, (i) testing of significance for difference of means of two populations with respect to a fuzzy set, (ii) testing of significance for difference of means of a population with respect to two fuzzy sets, (iii) to test the difference of variances of two populations with respect to a fuzzy set and (iv) to test the difference of variances of a population with respect to two fuzzy sets.The t-test statistic and F-test statistic are defined on the membership grades (MGs) of the fuzzy set over a random sample.The rules for taking decision about the hypotheses are provided.The optimistic and pessimistic approach, h-level set,   cut and fuzzy interval are not used in the proposed tests.The procedures of the proposed tests of hypotheses are illustrated by means of numerical examples.The proposed tests of hypotheses can help decision makers in the linguistic hypotheses tests related issues of real life problems by aiding them in the decision making process and providing an appropriate decision rules in an acceptable manner.

Preliminaries
The following concepts related to fuzzy set and its MF are used which can be found in George J. Klir and Bo Yuan (2008) and Chiang and Lin (1999).
Let X and Y be two crisp set and let A ~ and B ~ F be fuzzy sets where F is a fuzzy space.

If a fuzzy set
, then A ~ can be represented as follows: , then, they are written as follows: , then A ~ is written as follows: In Zadeh (1968), the probability of a fuzzy set A ~ defined on X with MF ) x ( A  is given by where P is the probability measure over X.From (1), we can conclude that the probability of the happening of the fuzzy event A ~ is the expectation of if the probability measure of X is known.Now, we need the following definitions of the sample mean and the sample variance of the MGs of a fuzzy set which can be found in Chiang and Lin (1999).
be a random sample of size n from a crisp set X with the MGs of a fuzzy set is defined as follows:

Testing of Significance for Difference of two Population Means with Respect to Fuzzy Sets
In this section, we propose the following two types of tests of statistical hypotheses: (i) Test for the difference of means of two populations using their small samples with respect to a fuzzy set.
(ii) Test for the difference of means of a population using its small sample with respect to two fuzzy sets.

Testing of Significance for the Difference of two Population Means with Respect to a Fuzzy Set
Let X and Y be two crisp population and A ~ be a fuzzy set defined on X and Y. Let { (one tailed test), the difference between (the means of populations with respect to A ~ are identical) at  level.Therefore, the NH is accepted.Otherwise, the AH ( corresponding to the given samples are given below: Now, with the help of the numerical example given below, the procedure of the above said testing of hypothesis is explained.
Example 3.1: Let X = {Students in a Government University } and Y = {Students in a Private University } be the two populations.Let the fuzzy set, A ~ = { comfort ability} be defined on X and Y. Now, we are going to test be the sample of size seven taken from the population Y.
Then, the membership grades of the given two samples based on their information concerning the fuzzy set A ~ are given below.
We take LOS, % 5   and the table value of t for 13 df at 5% LOS (one tailed test) is 2.16.Now, the test statistic, Therefore, the NH is rejected and the AH is accepted.Thus, on the basis of the sample, at 5% LOS, the government university students are more comfortable than private university students.

Testing of Significance for the Difference of Means of a Population with Respect to two Fuzzy Sets
Let X be a crisp population and A ~ and B ~ be two fuzzy sets defined on X.
} be a linguistic random sample of size m from a normal population with MGs If population standard deviations with respect to two fuzzy sets are the same, we use the test statistic for testing the NH, If population standard deviations with respect to two fuzzy sets are not the same, we use the test statistic for testing the NH, is, the mean of the population X with respect to A ~ and the mean of the population X with respect to B ~ are the same ( (two tailed test), the difference between That is, the means of the population X with respect to A ~ and B ~ are identical ( Therefore, the NH is accepted.Otherwise, the AH, that is, Now, the 100(1   )% confidence limits for the difference of the population means


corresponding to the given samples are given below: Therefore, NH is rejected and the AH is accepted, that is, the compassionate doctors need not be contentment in the city at 5% LOS.

Testing of Significance for Difference of two Variances with Respect to Fuzzy Sets
The following two types of tests of hypotheses are discussed in this section: (i) To test the difference of variances of two populations using their small samples with respect to a fuzzy set.
(ii) To test the difference of variances of one population using its small samples with respect to two fuzzy sets.Now, the test statistic

To
. Now, we have to test the hypothesis that the variance of X and the variance of Y are the same with respect to A ~, that is, the NH, o H : Let the LOS be  .

Now, the critical region of the AH, A
H for  LOS is given below: Alternative hypothesis Rejection region (one tailed test), the difference between the variances of X and Y with respect to A ~ at  level is not significant.That is, the population variances with respect to A ~ are identical (  level.Therefore, the NH is accepted.Otherwise, the AH, that is, (one tailed test), the difference between the variances of X and Y with respect to A ~ at  level is not significant.That is, the population variances with respect to (two tailed test), the difference between the variances of the population X and Y with respect to A ~ at  level is not significant.That is, the variance of the population X with respect to A ~ and the variance of the population Y with respect to A ~are the same ( ) at  level.Therefore, the NH is accepted.Otherwise, the AH, that is, Now, the 100(1   )% confidence limits for the quotient of variances  corresponding to the given samples are given below: Now, we explain the procedure of the above said hypothesis test with a numerical example.
Example 4.1: Let  X {All girls in a city} and  Y {All girls in a town}.Now, we test that the variability of the prettiness among girls in both places are the same.Now, a sample of six girls was taken at random from the city ) , , , , , ( = (Mary, Judy, Linda, Susan, Betty, Julia) and a sample of six girls was taken at random from the town ) , , , , , ( 6 5 4 3 2 1 y y y y y y = (Maya, Jasmine, Latha, Shela, Bindu, Jaya).Now, let us define a fuzzy set over the crisp set X and the crisp set Y , A ~={ Pretty girl}.Now, the membership grades of these two sets of six girls concerning the fuzzy set Now, the test statistic . Now, we have to test the hypothesis that the variances of the population X with respect to A ~ and B ~ are the same, that is, the NH, o H : Let the LOS be  .Now, for  LOS, the critical regions of the AH, A H of various types of tests are given below: Alternative hypothesis Critical region ) at  level.Therefore, the NH is accepted.Otherwise, the AH, that is, (one tailed test), the difference between the variances X with respect to A ~ and B ~ at  level is not significant.That is, the population variances with respect to A ~ and B ~ are identical ( ) at  level.Therefore, the NH is accepted.Otherwise, the AH, that is, (two tailed test), the difference between the variances of the population X with respect to A ~ and B ~ at  level is not significant.That is, the variance of the population X with respect to A ~ and the variance of the population X with respect to B ~ are the same (  corresponding to the given samples are given below:  Therefore, we accept the NH, that is, faculties in the University have no variability related to the behaviour of punctuality and availability.

Conclusion
Four types of tests of statistical hypotheses based on the MF of fuzzy sets which are totally different from conventional statistical hypothesis testing are proposed in this article.In the proposed tests of hypotheses, the differences of means and variances of the populations are studied with the help of fuzzy sets and small samples of the populations.The rules for decision taken about the hypotheses are provided.We can easily observe that the each proposed test of statistical hypothesis is a characteristic or attribute based test on the population.The optimistic and pessimistic approach, h-level set,   cut and fuzzy interval are not used in the proposed hypotheses tests.The proposed statistical hypotheses tests can help decision makers in tests of hypotheses related issues of real life problems for choosing an appropriate decision with satisfaction.
deviation of the MGs of fuzzy set A ~ over the random sample or the sample standard deviation of the MF of the fuzzy A 2,…,m over A ~ and B ~ respectively.Based on the sample, we test that the mean of the population X related to MGs of A of the population X related to MGs of B means of the MF of A ~ and B ~ defined on X be ) sample variances of the MF of A ~ and B ~ defined on X be test the NH, test), the difference between the variances of X with respect to A ~ and B ~ at  level is not significant.That, the population variances with respect to A ~ and B ~ are identical ( illustrate the procedure of the above said hypothesis test with help of a numerical example.Example 4.2: Let X = {Faculties in a University} be the population.Let ) of X.Let the fuzzy sets A ~ = {Punctuality} and B ~ = {Availability} be defined on X.Now, the MGs of the sample concerning fuzzy sets A ~ and B ~ based on the collected information are given below.
denote the table value of t for  df at  level.Now, for  LOS, the critical region of the alternative hypothesis (AH), A H is given below: be the table value of t for  df at  level.Now, for  LOS, the critical regions of the AH, A H for different types of tests are given below: t Now, the numerical example given below is used to illustrate the above said hypothesis test procedure..
.B ~={Contentment} be defined on X.We are going to test that in the city, compassionate doctors and contentment doctors are the same, that is, Test the Difference of Variances of two Populations with Respect to one Fuzzy Set Therefore, the NH is accepted at 5% level.Thus, city girls and town girls have the same degrees of variations of the prettiness, according to the random samples of girls.
We test the hypothesis that faculties in the University have no variability related to the behaviour of punctuality and availability.df at 5% level of significance, the table value of F is 3.78.