Solve the Problem $\sigma$ Or $\sigma ^ { 2 }$ , a Table Is Used to Find the Critical Values

Solve the problem.
-In constructing a confidence interval for $\sigma$ or $\sigma ^ { 2 }$ , a table is used to find the critical values $\chi _ { L } ^ { 2 }$ and $\chi _ { R } ^ { 2 }$ for values
of $n \leq 101$ . For larger values of $n , \chi _ { L } ^ { 2 }$ and $\chi \frac { 2 } { R }$ can be approximated by using the following formula:
$$\chi ^ { 2 } = \frac { 1 } { 2 } \left[ \pm z _ { \alpha / 2 } + \sqrt { 2 \mathrm { k } - 1 } \right] ^ { 2 }$$
where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z } _ { \alpha / 2 }$ is the critical $\mathrm { z }$ score. Estimate the critical values $\chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ for a situation in which you wish to construct a $95 \%$ confidence interval for $\sigma$ and in which the sample size is $\mathrm { n } = 298$ .

A) $\chi _ { \mathrm { L } } ^ { 2 } \approx 258.727 , \chi _ { \mathrm { R } } ^ { 2 } \approx 338.979$

B) $\chi _ { L } ^ { 2 } \approx 250.692 , \chi _ { R } ^ { 2 } \approx 346.150$

C) $\chi _ { \mathrm { L } } ^ { 2 } \approx 251.611 , \chi _ { \mathrm { R } } ^ { 2 } \approx 347.230$

D) $x _ { \mathrm { L } } ^ { 2 } \approx 257.795 , \chi _ { R } ^ { 2 } \approx 337.911$