# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 4 Possibilistic Statistical Tests of Crisp Hypotheses
Hypotheses Tests for the Variance of a Normal Distribution**  
We test at significance level \(\gamma\) the null hypothesis \(H_{0}:\ \sigma^{2}=\sigma_{0}^{2}\) for the variance \(\sigma^{2}\) of a normal random variable \(X\), using a random sample of observations of \(X\) of size \(n\) and variance \(s^{2}\). In the crisp case, we test \(H_{0}\), using the test statistic  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]3  
where \(S^{2}\) is the sample variance, which follows \(\chi^{2}_{n-1}\) distribution with \(n-1\) degrees of freedom under the null hypothesis [1]. \(H_{0}\) is rejected from a given random sample (omitting in \(\chi^{2}\) the implied \(n_{1}-1\) degrees of freedom):  
(a) For _one-sided test from the left_ (alternative \(H_{1}:\ \sigma^{2}<\sigma_{0}^{2}\)), if  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]4  
and \(F^{-1}\) is the inverse distribution function of the \(\chi^{2}_{n-1}\) distribution. (b) For _one-sided test from the right_ (alternative \(H_{1}:\ \sigma^{2}>\sigma_{0}^{2}\)), if  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]5  
(c) For _two-sided_ test (alternative \(H_{1}:\ \sigma^{2}\
eq\sigma_{0}^{2}\)), if  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]6  
where  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]7  
and  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]8  
the crisp value of the statistic (43). Otherwise, \(H_{0}\) is not rejected. In the fuzzy case for the test of \(H_{0}\), we use the fuzzy statistic \(\widetilde{\chi^{2}}\) generated by substituting \(S^{2}\) in (43) with a fuzzy estimator \(\widetilde{s}^{2}\) of the following variance:  
\[z_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{42}\]9  
Using the \(\alpha-\)cuts of the unbiased fuzzy estimator given in (14) and interval arithmetics, we obtain from (48) the \(\alpha-\)cuts of the possibility distribution of the following test statistic:  
\[z_{0}=\frac{17.61-20.2}{8.96/\sqrt{50}}=-2.044\]0  
As illustrated in Figures 8 and 9, the shape of this test statistic is significantly different than the shape of the respective test statistic produced by the implementation of \(\mu_{\widetilde{\omega_{2}^{2}}}\) used in [29], which is as expected since the shape of the two implementations (14) and (18) of \(\mu_{\widetilde{\omega_{2}^{2}}}\) are different. \[z_{0}=\frac{17.61-20.2}{8.96/\sqrt{50}}=-2.044\]1 The possibility distribution of the fuzzy statistic \(\widetilde{\chi^{2}}[\alpha]\) for the test of \(H_{0}\) of Example 3. \[z_{0}=\frac{17.61-20.2}{8.96/\sqrt{50}}=-2.044\]2 The possibility distribution of the fuzzy statistic \(\widetilde{\chi^{2}}[\alpha]\) and the fuzzy critical values for the respective test of \(H_{0}\) of Example 3 in [29]. The membership functions of the critical values \(\widetilde{CV}_{1}\) and \(\widetilde{CV}_{2}\) of one-sided tests are as follows:  
\[\mu_{\widetilde{CV}_{1}}(x)=\left\{ \begin{array}{ll}1,&x=F^{-1}(1-\gamma)\\ 0,&\text{elsewhere}\end{array} \right.\quad\text{and}\;\;\mu_{\widetilde{CV}_{ 2}}(x)=\left\{ \begin{array}{ll}1,&x=F^{-1}(\gamma)\\ 0,&\text{elsewhere}\end{array} \right. \tag{50}\]  
and for the two-sided test  
\[\mu_{\widetilde{CV}_{1}}(x)=\left\{ \begin{array}{ll}1,&x=F^{-1}\Big{(}1- \frac{\gamma}{2}\Big{)}\\ 0,&\text{elsewhere}\end{array} \right.\quad\text{and}\;\;\mu_{\widetilde{CV}_{ 2}}(x)=\left\{ \begin{array}{ll}1,&x=F^{-1}\Big{(}\frac{\gamma}{2}\Big{)}\\ 0,&\text{elsewhere}\end{array} \right. \tag{51}\]  
where \(F^{-1}\) the inverse distribution function of the \(\chi^{2}\) distribution with \(n-1\) degrees of freedom. Having the \(\alpha-\)cuts of the possibility distribution of the fuzzy statistic \(\chi^{2}_{0}\) and the critical values \(\widetilde{CV}_{1}\) and \(\widetilde{CV}_{2}\), we can evaluate the possibility of rejection or acceptance of \(H_{0}\) from (21), (23), (26), (28) for \(\widetilde{U}=\widetilde{\chi^{2}}\) and (50) for one sided tests or from (31), (32), (34) and (51) for two sided test. **Example 3**.