# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 7 Hypothesis Tests for the Mean of a Normal Random Variable with Unknown Variance
Having the \(\alpha-\)cuts (57) of the possibility distribution of the fuzzy statistic \(T\) 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 \(\widetilde{U}=\widetilde{T}\) and (58) for one-sided tests or from (31), (32), (34) and (59) for the two-sided test. **Example 5**. : _We test the null hypothesis \(H_{0}:\mu=11\) at significance level \(0.02\) with alternative \(H_{1}:\mu\
eq 11\) (two sided test) for the mean \(\mu\) of a normal random variable \(X\), using a sample of \(n=25\) observations with sample mean and variance \(\overline{x}=10,9\) and \(s^{2}=7.57\)._  
In the crisp case, the value of the statistic (52) is evaluated by (53)  
\[t_{0}=\frac{10,9-11}{\sqrt{3,57}/\sqrt{25}}=0,26\]  
so the \(p-\)value is \(p=2(1-F_{24}(0,26))=0.8\), which is much higher than the significance level \(\gamma=0.02\). Hence, \(H_{0}\) is accepted by the crisp test with high \(p\)-value (the test statistic is almost at the center of the acceptance region). We apply the above two-sided fuzzy test of \(H_{0}\), implementing (57) and (59). So, we obtain the results of Figure 13, where the core of \(\widetilde{T}\) is between the cores of \(\widetilde{CV}_{1}\) and \(\widetilde{CV}_{2}\) and the points of intersection of \(\mu_{\widetilde{T}}\) and \(\mu_{\widetilde{CV}_{1}}\) and of \(\mu_{\widetilde{T}}\) and \(\mu_{\widetilde{CV}_{2}}\) have \(y_{1}=0.07\) and \(y_{2}=0.05\). Hence, according to (34), the possibility of acceptance of \(H_{0}\) is  
\[\varphi_{A}=\min(1-y_{1},1-y_{2})=0.93,\]  
so \(H_{0}\) is accepted by this test with possibility \(d=0.93\). Applying the respective test of [29; 30], we get the results of Figure 14, where we see that the possibility of acceptance of \(H_{0}\) is as follows:  
\[\varphi_{A}=\min(1-y_{1},1-y_{2})=0.69\]  
Figure 13: The possibility distribution of the fuzzy statistic \(\overline{T}\) for the test of \(H_{0}\) of Example 5. **Example 6**. : _We test the null hypothesis \(H_{0}:\mu=6\) at significance level \(0.05\) with alternative \(H_{1}:\mu\
eq 6\) (two sided test) for the mean \(\mu\) of a normal random variable \(X\), using a sample of \(n=25\) observations with sample mean and variance \(\overline{x}=4.5\) and \(s^{2}=3.66\)._  
In the crisp case, the value of the statistic (52) is evaluated by (53)  
\[t_{0}=\frac{4.5-6}{\sqrt{3,66}/\sqrt{25}}=-3.92\]  
so the \(p-\)value is \(p=2F_{24}(-3.92))=6.5\times 10^{-4}\), which is almost zero. Hence, \(H_{0}\) is rejected by the crisp test. We apply the above test implementing (57) and (59). So, we take the results of Figure 15, where the core of \(\overline{T}\) is at the left of the core of \(\overline{CV}_{2}\) and the point of intersection of \(\overline{T}\) and \(\overline{CV}_{2}\) has \(y_{2}=0.03\). Hence, according to (32), the possibility of rejection of \(H_{0}\) is \(1-0.0.3=0.97\). Applying the respective test of [30], we get the results of Figure 16, where we see that the possibility of acceptance of \(H_{0}\) is as follows:  
\[\varphi_{A}=\min(1-y_{1},1-y_{2})=1-0.33=0.67\]  
Figure 14: The possibility distribution of the fuzzy statistic \(\overline{T}\) and the fuzzy critical values for the respective test of [15] of \(H_{0}\) of Example 5. Figure 15: The fuzzy statistic \(\overline{T}\) for the test of Example 6. A similar result with even lower possibility of acceptance is obtained by the respective test of [29].