# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 4 Possibilistic Statistical Tests of Crisp Hypotheses
: _We test the null hypothesis \(H_{0}\): \(\mu=20.2\) at significance level \(\gamma=0.05\) with alternative \(H_{1}\): \(\mu\
eq 20.2\) (two-sided test) for the mean value \(\mu\) of the temperature \(X\), using the two large random samples of 50 observations each of the Appendix A (monthly values for selected Greek weather stations) [37]._  
The mean and the standard deviation of the first sample are \(\overline{x}_{1}=17.61\) and \(s_{1}=8.96\), so in the crisp test, we evaluate the value of the statistic \(Z\) for the first sample from (42):  
\[z_{0}=\frac{17.61-20.2}{8.96/\sqrt{50}}=-2.044\]  
Since  
\[z_{0}=-2.044<-z_{0.05/2}=-z_{0.025}=-1.96\]  
\(H_{0}\) is rejected by this crisp test. For the second sample the mean and variance are \(\overline{x}_{2}=17.95\) and \(s_{2}=8.51\), so the value of the statistic \(Z\) in this case is the following:  
\[z_{0}=\frac{17.95-20.2}{8.51/\sqrt{50}}=-1.87\]  
Therefore, since the following is true:  
\[z_{0}=-1.87>-z_{0.05/2}=-z_{0.025}=-1.96\]  
\(H_{0}\) is not rejected by the crisp test. Applying the above described two-sided fuzzy test of \(H_{0}\), implementing the possibility distribution (39) of \(\widetilde{Z}_{0}\) for the value \(z_{0}\) of (42) and the critical values (41) for the first sample, we get the results of Figure 4, where the point of intersection of \(\mu_{Z_{0}}\) and \(\mu_{\widetilde{CV}_{2}}\) has \(y_{0}=0.94\). So according to (32), the possibility of rejection of \(H_{0}\) is \(1-0.94=0.06\). Hence, we cannot make a decision on whether to reject or not \(H_{0}\) from this sample. Implementing (39) and (41) for the second sample, we take Figure 5, where the point of intersection of \(\mu_{Z_{0}}\) and \(\mu_{\widetilde{CV}_{2}}\) has \(y_{0}=0.95\), so according to (34), the possibility of acceptance of \(H_{0}\) is \(1-0.95=0.05\). Therefore, we cannot make a decision on whether to reject or not \(H_{0}\) from this sample. Figure 4: The possibility distribution of the fuzzy statistic \(Z\) for the test of Example 1 from a sample with \(\overline{x}_{1}=17.6\). Figure 5: The possibility distribution of the fuzzy statistic \(Z\) for the test of Example 1 from a sample with \(\overline{x}_{2}=17.95\). **Example 2**. : _We test at significance level \(0.05\) the null hypothesis \(H_{0}\): \(\mu=11\) with alternative \(H_{1}\): \(\mu\
eq 11\) (two-sided test) for the mean \(\mu\) of a random variable \(X\), which follows normal distribution with variance \(\sigma^{2}=10.24\), using a sample of \(n=25\) observations with sample mean \(\overline{x}=11.05\)._  
The value of the test statistic (35) is found by (36) to be as follows:  
\[z_{0}=\frac{11.05-11}{\sqrt{10.24/\sqrt{25}}}=0.08\]  
so the crisp \(p\)-value of the test is as follows:  
\[p=2(1-\Phi(0.08))=0.94\]  
Therefore, in this case, the crisp test gives acceptance of \(H_{0}\) with a very high \(p\)-value (near to the maximum value, 1). We apply the above two-sided fuzzy test of \(H_{0}\) implementing (39) and (41). So, we obtain the results of Figure 6, where the core of \(\widetilde{Z}\) is between the cores of \(\widetilde{CV}_{1}\) and the point of intersection of \(\mu_{\widetilde{Z}}\), and \(\mu_{\widetilde{CV}_{1}}\) has \(y_{1}=0.04\) and of \(\mu_{\widetilde{Z}}\) and \(\mu_{\widetilde{CV}_{2}}\) has \(y_{2}=0.06\). Hence, according to (34), the possibility of acceptance of \(H_{0}\) is as follows:  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]0  
so \(H_{0}\) is accepted by this test with possibility \(d=0.94\). Applying the respective test of [29; 30], we get the results of Figure 7, where we see that the possibility of acceptance of \(H_{0}\) is much lower than 1:  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]1  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]2 The possibility distribution of the fuzzy statistic \(\widetilde{Z}\) for the two-sided test of Example 2. **6.