Chapter 7.3, Problem 23BB

Finding Critical Values In constructing confidence intervals for σ or σ², 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

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

where k is the number of degrees of freedom and z_α/2     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?