Chapter 8.4, Problem 20BB

Finding Critical Values of χ² Repeat Exercise 19 using this approximation (with k and z as described in Exercise 19):

${\chi }^2=k{\left(1-\frac{2}{9k}+z\sqrt{\frac{2}{9k}}\right)}^3$

19. Finding Critical Values of χ² For large numbers of degrees of freedom, we can approximate critical values of χ² as follows:

${\chi }^2=\frac{1}{2}{\left(z+\sqrt{2k-1}\right)}^2$

Here k is the number of degrees of freedom and z is the critical value(s) found from technology or Table A-2. In Exercise 12 “Spoken Words” we have df = 55, so Table A-4 does not list an exact critical value. If we want to approximate a critical value of χ² in the right-tailed hypothesis test with α = 0.01 and a sample size of 56, we let k = 55 with z = 2.33 (or the more accurate value of z = 2.326348 found from technology). Use this approximation to estimate the critical value of χ² for Exercise 12. How close is it to the critical value of χ² = 82.292 obtained by using Statdisk and Minitab?