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 $\mathrm { n } \leq 101$ . For larger values of $n , \chi _ { L } ^ { 2 }$ and $\chi \frac { 2 } { \mathrm { R } }$ 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. Estimate the critical values $x _ { L } ^ { 2 }$ and $x _ { R } ^ { 2 }$ for a situation in which you wish to construct a $95 \%$ confidence interval for $\sigma$ and in which the sample size is $n = 232$ .

A) $\chi _ { L } ^ { 2 } \approx 190.338 , \chi _ { R } ^ { 2 } \approx 274.504$
B) $\chi _ { L } ^ { 2 } \approx 191.247 , \chi _ { R } ^ { 2 } \approx 275.595$
C) $\chi _ { L } ^ { 2 } \approx 196.533 , \chi _ { \mathrm { R } } ^ { 2 } \approx 267.173$
D) $\chi \underset { L } { 2 } \approx 197.457 , \chi \underset { R } { 2 } \approx 268.249$