?Finding Critical Values In constructing confidence intervals for \(\sigma\) or

Chapter 7, Problem 23

(choose chapter or problem)

Finding Critical Values In constructing confidence intervals for \(\sigma\) or \(\sigma^{2}\), Table A-4 can be used to find the critical values \(\chi_{L}^{2}\) and \(\chi_{R}^{2}\) only for select values of n up to 101, so the number of degrees of freedom is 100 or smaller. For larger numbers of degrees of freedom, we can approximate \(\chi_{L}^{2}\) and \(\chi_{R}^{2}\) by using

\(\chi^{2}=\frac{1}{2}\left[\pm z_{\alpha / 2}+\sqrt{2 k-1}\right]^{2}\)

where k is the number of degrees of freedom and \(z_{\alpha / 2}\) is the critical z score described in Section 7-1. Use this approximation to find the critical values \(\chi_{L}^{2}\) and \(\chi_{R}^{2}\) for Exercise 8 “Heights of Men,” where the sample size is 153 and the confidence level is 99%. How do the results compare to the actual critical values of \(\chi_{L}^{2}=110.846\) and \(\chi_{R}^{2}=200.657^{\circ}\)?

Text Transcription:

\sigma

\sigma^2

\chi_L^2

\chi_R^2

\chi^2=\frac12\[\pm z_\alpha / 2+\sqrt2 k-1\^2

z_\alpha / 2

\chi_L^2=110.846

\chi_R^2=200.657^\