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 $\mathrm { n } \leq 101$ . For larger values of $\mathrm { n } , \chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ can be approximated by using the following formula:
$$\chi ^ { 2 } = \frac { 1 } { 2 } \left[ \pm z _ { \alpha / 2 } + \sqrt { 2 k - 1 } \right] ^ { 2 }$$
where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z } \alpha / 2$ is the critical z score. Construct the $90 \%$ confidence interv: for $\sigma$ using the following sample data: a sample of size $n = 261$ yields a mean weight of $154 \mathrm { lb }$ and a standard deviation of $25.3 \mathrm { lb }$ . Round the confidence interval limits to the nearest hundredth.

A) $23.58 < \sigma < 27.24$
B) $23.98 < \sigma < 26.83$
C) $23.32 < \sigma < 27.71$
D) $23.62 < \sigma < 27.30$