Reply to "open problem of fuzzy confidence interval for fuzzy process capability index"

The recent paper by Parchami et al. [2] proposes an open problem: Let $\ \hat{\tilde{C\!}}_p = T\left {\frac{{a_u-c_l }} {{6s}},\frac{{b_u-b_l }} {{6s}},\frac{{c_u-a_l }} {{6s}}} \right\$ be a point estimate of fuzzy process capability index $\\tilde{C}_p\$ as definition of Parchami et al. [2], where $\ s = \sqrt {\frac{1}{{n-1}}\sum\limits_{i = 1}^n {\left {x_i-\bar x} \right^2}}\$. Is it true that: $\ \mathop {\lim }\limits_{n \to \infty } \left[ {\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n-1,\alpha /2}^2 }} {{n-1}}},\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n-1,1-\alpha /2}^2 }} {{n-1}}}} \right] = \left\{ {\hat{\tilde{C\!}}_p } \right\}?\$ We modify their open problem and prove that “$\ \mathop {\lim }\limits_{n \to \infty } \left[ {\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n-1,\alpha /2}^2 }} {{n-1}}},\hat{\tilde{C\!}}_{p \otimes } \sqrt {\frac{{\chi _{n-1,1-\alpha /2}^2 }} {{n-1}}} } \right] \cong \left\{ {\hat{\tilde{C\!}}_p } \right\}\$ {for large} n” is true.