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 $\chi { } _ { L } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ for values of $\mathrm { n } \leq 101$ . For larger values of $n , \chi _ { \mathrm { L } } ^ { 2 }$ and $\chi _ { \mathrm { R } } ^ { 2 }$ can be approximated by using the following formula: $\chi ^ { 2 } = \frac { 1 } { 2 } \left[ \pm \mathrm { z } _ { \alpha / 2 } + \sqrt { 2 \mathrm { k } - 1 } \right] ^ { 2 }$ where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z } _ { \alpha / 2 }$ is the critical 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 $n = 295$ .

A) $\chi _ { L } ^ { 2 } \approx 255.930 , \chi _ { R } ^ { 2 } \approx 335.776$

B) $\chi _ { \mathrm { L } } ^ { 2 } \approx 247.934 , \chi _ { \mathrm { R } } ^ { 2 } \approx 342.908$

C) $\chi _ { L } ^ { 2 } \approx 248.853 , \chi _ { R } ^ { 2 } \approx 343.989$

D) $\chi _ { \mathrm { L } } ^ { 2 } \approx 254.998 , \chi _ { \mathrm { R } } ^ { 2 } \approx 334.708$