In Constructing a Confidence Interval For $\sigma$ Or $\sigma ^ { 2 }$

In constructing a confidence interval for $\sigma$ or $\sigma ^ { 2 }$ , a table is used to find the critical values $\chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ for values of $n \leq 101$ . For larger values of $n , \chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ can be approximated by using the following formula: $\chi ^ { 2 } = \frac { 1 } { 2 } \pm z _ { \alpha / 2 } + \sqrt { 2 k - 1 } ^ { 2 }$ where $k$ is the number of degrees of freedom and $z _ { \alpha / 2 }$ is the critical $z$ -score. Construct the $90 \%$ confidence interval for $\sigma$ using the following sample data: a sample of size $n = 232$ yields a mean weight of $154 \mathrm { lb }$ and a standard deviation of $25.5 \mathrm { lb }$ . Round the confidence interval limits to the nearest hundredth.

A) $23.66 \mathrm { lb } < \sigma < 27.58 \mathrm { lb }$
B) $23.39 \mathrm { lb } < \sigma < 28.09 \mathrm { lb }$
C) $24.09 \mathrm { lb } < \sigma < 27.15 \mathrm { lb }$
D) $23.71 \mathrm { lb } < \sigma < 27.65 \mathrm { lb }$