Provide an Appropriate Response $\sigma$ Or $\sigma ^ { 2 }$ , a Table Is Used to Find the Critical Values

Multiple Choice

Provide an appropriate response.
-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: $\chi ^ { 2 } = \frac { 1 } { 2 }$ $\left[ \pm \mathrm { z } _ { \alpha / 2 } + \sqrt { 2 \mathrm { k } - 1 } \right] ^ { 2 }$ where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z } _ { \alpha / 2 }$ is the critical $\mathrm { z }$ score. 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) $23.71 \mathrm { lb } < \sigma < 27.65 \mathrm { lb }$
B) $23.66 \mathrm { lb } < \sigma < 27.58 \mathrm { lb }$
C) $23.39 \mathrm { lb } < \sigma < 28.09 \mathrm { lb }$
D) $24.09 \mathrm { lb } < \sigma < 27.15 \mathrm { lb }$