# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 7 Hypothesis Tests for the Mean of a Normal Random Variable with Unknown Variance  
We test at significance level \(\gamma\) the null hypothesis \(H_{0}:\mu=\mu_{0}\) for the mean value \(\mu\) of a random variable \(X\), which follows normal distribution with unknown variance, using a random sample of observations of \(X\) of size \(n\) with sample mean and variance \(\overline{x}\) and \(s^{2}\). In this case, the test statistic is  
\[T=\frac{\overline{X}-\mu_{0}}{S/\sqrt{n}} \tag{52}\]  
where \(\overline{X}\) and \(S\) are the statistics of the sample mean and the standard deviation. It is known [1] that under the null hypothesis (\(\mu=\mu_{0}\)), \(T\) follows \(t\) distribution with \(n-1\) degrees of freedom. So in the crisp case, \(H_{0}\) is rejected from a given random sample: (a) For the _one sided test from the right_ (alternative \(H_{1}:\mu>\mu_{0}\)), if \(t_{0}>t_{\gamma;n-1}\), where  
\[t_{0}=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}} \tag{53}\]  
the value of the statistic (52),  
\[t_{\gamma;n-1}=F_{n-1}^{-1}(1-\gamma) \tag{54}\]  
the critical value of the test and \(F_{n-1}^{-1}\) is the inverse distribution function of the \(t\) distribution with \(n-1\) degrees of freedom. (b) For the _one sided test from the left_ (alternative \(H_{1}:\mu<\mu_{0}\)), if \(t_{0}<-t_{\gamma;n-1}\). (c) For the _two-sided_ test ( alternative \(H_{1}:\mu\
eq\mu_{0}\)), if \(t_{0}<-t_{\gamma/2;n-1}\) or \(t_{0}>t_{\gamma/2;n-1}\) while, if \(-t_{\gamma/2;n-1}<t_{0}<t_{\gamma/2;n-1}\), then \(H_{0}\) is not rejected. In the fuzzy case for the test of \(H_{0}\) we use the following fuzzy statistic:  
\[\overline{T}=\frac{\overline{\mu}-\mu_{0}}{\overline{s}/\sqrt{n}} \tag{55}\]  
which is generated by substituting \(\overline{X}\) and \(S\) in (52) with the fuzzy estimators \(\widetilde{\mu}\) and \(\widetilde{s}\) of the mean and standard deviation. The \(\alpha-\)cuts of \(\widetilde{s}\) are given by (13) and (as described in [29; 30]) the \(\alpha-\)cuts of the possibility distribution of \(\widetilde{\mu}\) are as follows:  
\[\mu_{\widetilde{\mu}}[\alpha]=\left[\overline{x}-t_{\alpha;n-1}\frac{s}{ \sqrt{n}},\ \overline{x}+t_{\alpha;n-1}\frac{s}{\sqrt{n}}\right],\ \ \ \ \ \ \alpha\in[0,1] \tag{56}\]  
From (53), (55), (56), (13) and fuzzy number arithmetics follows that the \(\alpha-\)cuts of the possibility distribution of the fuzzy statistic \(\widetilde{T}\) are as follows:  
\[\mu_{\widetilde{T}}[\alpha] = \left[\frac{\overline{x}-\mu_{0}-t_{\alpha;n-1}\frac{s}{\sqrt{n} }}{\frac{1}{\sqrt{n}}s\sqrt{\frac{(n-1)}{\chi^{2}_{1-\alpha;n-1}}}}\,\ \ \ \frac{\overline{x}-\mu_{0}+t_{\alpha;n-1}\frac{s}{\sqrt{n}}}{ \frac{1}{\sqrt{n}}s\sqrt{\frac{(n-1)}{\chi^{2}_{\alpha;n-1}}}}\right] \tag{57}\] \[= \left[\sqrt{\frac{\chi^{2}_{1-\alpha;n-1}}{n-1}}(t_{0}-t_{\alpha; n-1})\,\ \ \sqrt{\frac{\chi^{2}_{\alpha;n-1}}{n-1}}(t_{0}+t_{\alpha;n-1})\right]\]  
where \(t_{\alpha}=F_{n-1}^{-1}(1-\alpha)\). The membership functions of the critical values \(\widetilde{CV}_{1}\) and \(\widetilde{CV}_{2}\) of the one-sided tests are as follows:  
\[\mu_{\widetilde{CV}_{1}}(x)=\left\{ \begin{array}{ll}1,&x=F_{n-1}^{-1}(1- \gamma)\\ 0,&\mbox{elsewhere}\end{array} \right.\ \ \ \mbox{and}\ \ \ \mu_{\widetilde{CV}_{2}}(x)=\left\{ \begin{array}{ll}1,&x=F_{n-1}^{-1}(\gamma)\\ 0,&\mbox{elsewhere}\end{array} \right. \tag{58}\]and for the two-sided test,  
\[\mu_{\widetilde{CV}_{1}}(x)=\left\{ \begin{array}{ll}1,&x=F_{n-1}^{-1}\Big{(}1- \frac{\gamma}{2}\Big{)}\\ 0,&\text{elsewhere}\end{array} \right.\quad\text{and}\;\;\mu_{\widetilde{CV}_{2 }}(x)=\left\{ \begin{array}{ll}1,&x=F_{n-1}^{-1}\Big{(}\frac{\gamma}{2} \Big{)}\\ 0,&\text{elsewhere}\end{array} \right. \tag{59}\]  
where \(F^{-1}\) is the inverse distribution function of the \(t\) distribution with \(n-1\) degrees of freedom.