# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 4 Possibilistic Statistical Tests of Crisp Hypotheses
: _We test at significance level \(0.01\) the null hypothesis \(H_{0}:\;\sigma^{2}=2\) with alternative \(H_{1}:\;\sigma^{2}\
eq 2\) for the variance of a normal random variable, using a random sample of \(101\) observations with sample variance \(s^{2}=2\)._  
The crisp test in this case gives acceptance of \(H_{0}\) with the largest possible difference between the test statistic and the critical values since the \(p\)-value is \(1\) (the test statistic is exactly in the middle of the no-rejection region), so it is the best case for the acceptance of \(H_{0}\). Applying the above fuzzy test, we get the results of Figure 8, where the core of \(\widetilde{\chi^{2}}\) is between the cores of \(\widetilde{CV}_{1}\) and \(\widetilde{CV}_{2}\) and the point of intersection of \(\mu_{\widetilde{\chi^{2}}}\), and \(\;\mu_{\widetilde{CV}_{1}}\) has \(y_{1}=0.06\) and of \(\mu_{\widetilde{\chi^{2}}}\) and \(\mu_{\widetilde{CV}_{2}}\), \(y_{2}=0.03\), so according to (34) the possibility of acceptance of \(H_{0}\) is as follows:  
\[\varphi_{A}=\text{Ness}\Big{(}\widetilde{\chi^{2}}<\widetilde{CV}_{1},\; \widetilde{\chi^{2}}>\widetilde{CV}_{2}\Big{)}=\min(1-0.06,1-0.03)=0.94.\]  
The respective fuzzy test of \(H_{0}\) of [29] gives the results of Figure 9, where the possibility of acceptance of \(H_{0}\) is as follows:  
\[\varphi_{A}=\text{[PERSON]}\Big{(}\widetilde{\chi^{2}}<\widetilde{CV}_{1},\; \widetilde{\chi^{2}}>\widetilde{CV}_{2}\Big{)}=\min(1-y_{1},1-y_{2})=0.44.\]  
and the test of \(H_{0}\) of [30] gives the results of Figure 10, where the possibility of acceptance of \(H_{0}\) is as follows:  
\[\varphi_{A}=\text{[PERSON]}\Big{(}\widetilde{\chi^{2}}<\widetilde{CV}_{1},\; \widetilde{\chi^{2}}>\widetilde{CV}_{2}\Big{)}=\min(1-0.17,1-0.23)=0.77.\]The results of the fuzzy hypothesis tests proposed in this paper are in much better agreement with the results of the respective crisp tests than the results of the respective tests of [29]. This happens because of the following:  
1. The former are based on the correct implementation of \(\widetilde{\sigma_{u}^{2}}\), according to the theory, whereas the latter contain the error of using \(\lambda\)-cuts instead of \(\alpha\)-cuts. 2. The tests of [29] use fuzzy critical values, which are created using probabilistic concepts. The above are illustrated by the characteristic case of the hypothesis test of Example 3 in which the crisp value of the test statistic is exactly in the middle of the acceptance region (\(p\)-value \(\,=\,1\)), which is the best case of acceptance. As shown in this example, the test, which uses the implementation of \(\widetilde{\sigma_{u}^{2}}\) of [29], gives a significantly lower possibility of acceptance of \(H_{0}\) (0.44) than the test which uses our implementation of (0.94). So, the latter is in much better agreement with the crisp statistical tests than the former. The possibilistic tests developed in this paper are in better agreement with the results of the respective crisp tests than the results of the tests of [30], which also use the same fuzzy critical values as [29]. This is illustrated in Examples 2 and 3 in which the crisp value of the test statistic is exactly in the middle of the acceptance region. As shown in these examples, the tests of [30] give a significantly lower degree of acceptance of \(H_{0}\) (0.69 for Example 2 and 0.77 for Example 3) than the respective tests of this paper (0.94 for both examples). So, the latter are in better agreement with the crisp statistical test than the former. This happens in all relevant examples (statistic in the middle of the acceptance region). **Example 4**.