# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 4 Possibilistic Statistical Tests of Crisp Hypotheses
It is known that under the null hypothesis (\(\mu=\mu_{0}\)) \(Z\) follows the standard normal distribution \(N(0,1)\), so \(H_{0}\) is rejected from a given sample:  
(a) For the _one sided test from the right_, if  
\[z_{0}\geq z_{\gamma},\ \ \ \text{where}\ \ \ z_{\gamma}=\Phi^{-1}(1-\gamma),\]  
(b) For the _one sided test from the left_, if \(z_{0}\leq-z_{\gamma}=\Phi^{-1}(\gamma)\),  
(c) For the _two sided_ test, if \(z_{0}\leq-z_{\gamma/2}\) or \(z_{0}\geq z_{\gamma/2}\), where  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{\sigma/\sqrt{n}} \tag{36}\]  
the value of the statistic (35) for the given sample,  
\[z_{\gamma/2}=\Phi^{-1}\Big{(}1-\frac{\gamma}{2}\Big{)} \tag{37}\]  
and \(\Phi^{-1}\) the inverse distribution function of the standard normal distribution. Meanwhile, \(H_{0}\) is not rejected if \(-z_{\gamma/2}<z_{0}<z_{\gamma/2}\). In the fuzzy case, the test of \(H_{0}\) is based on the fuzzy statistic  
\[\widetilde{Z}=\frac{\widetilde{\mu}-\mu_{0}}{\sigma/\sqrt{n}}, \tag{38}\]  
which is generated by substituting \(\overline{X}\) in (35) with the fuzzy estimator \(\widetilde{\mu}\) of the mean value for the given sample, the \(\alpha-\)cuts of the membership function of which are given by (7). From (38), (7) and interval arithmetics follow that the \(\alpha-\)cuts of the possibility distribution of the fuzzy statistic \(\widetilde{Z}\) are as follows:  
\[\mu_{\widetilde{Z}}[\alpha]=[Z-z_{\alpha},\ Z+z_{\alpha}],\ \ \ \alpha\in[0,1]\]  
Hence, the \(\alpha-\)cuts of the possibility distribution of \(\widetilde{Z}\) for the given sample are as follows:  
\[\varphi_{R}(X_{1},X_{2},\ldots,X_{n})=1-y_{2} \tag{32}\]0  
where \(z_{0}\) the crisp value of the statistic \(Z\) for this sample, which is given by (36). The membership functions of the critical values \(\widetilde{CV_{1}}\) and \(\widetilde{CV_{2}}\) of the one-sided tests are as follows:  
\[\mu_{\widetilde{CV_{1}}}(z)=\left\{ \begin{array}{ll}1,&z=\Phi^{-1}(1-\gamma )\\ 0,&\text{elsewhere}\end{array} \right.\ \ \text{and}\ \ \mu_{\widetilde{CV_{2}}}(z)= \left\{ \begin{array}{ll}1,&z=\Phi^{-1}(\gamma)\\ 0,&\text{elsewhere}\end{array} \right. \tag{40}\]  
and for the two-sided test,  
\[\mu_{\widetilde{CV_{1}}}(z)=\left\{ \begin{array}{ll}1,&z=\Phi^{-1}\big{(}1 -\frac{\gamma}{2}\big{)}\\ 0,&\text{elsewhere}\end{array} \right.\ \ \text{and}\ \ \mu_{\widetilde{CV_{2}}}(z)= \left\{ \begin{array}{ll}1,&z=\Phi^{-1}\big{(}\frac{\gamma}{2}\big{)}\\ 0,&\text{elsewhere}\end{array} \right. \tag{41}\]  
Having the \(\alpha-\)cuts of the possibility distribution of the fuzzy statistic \(\widetilde{Z}_{0}\) of (39) and the critical values \(\widetilde{CV_{1}}\) and \(\widetilde{CV_{2}}\), we can evaluate the possibility of the rejection or acceptance of \(H_{0}\) from (21), (23), (26), (28) for \(\widetilde{U}=\widetilde{Z}_{0}\) and (40) for one sided tests or from (31), (32), (34) and (41) for the two-sided test. If the sample is large and \(X\) follows any distribution, then in the crisp case, we test \(H_{0}\) using the statistic  
\[Z=\frac{\overline{X}-\mu_{0}}{s/\sqrt{n}}\]  
where \(\overline{X}\) is the statistic of the sample mean and \(s\), the sample standard deviation [1]. It is known that under the null hypothesis \((\mu=\mu_{0})\), \(Z\) follows the standard normal distribution \(N(0,1)\) according to the central limit theorem [1], so \(H_{0}\) is rejected or not by one sided tests, using (21), (23), (26), (28) for \(\widetilde{U}=\widetilde{Z}_{0}\) given in (39) for the following sample value:  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]  
of the statistic \(Z\) or (31), (32), (34) for the two-sided test. **Example 1**.