Couples were recruited for a study of how many words people speak in a day. A random sample of 58 males resulted in a mean of 14,602 words and a standard deviation of 7807 words. Use a 0.05 significance level to test the claim that males have a standard deviation that is greater than the standard deviation of 7460 words for females. Use the accompanying method of approximation to estimate the critical value of $\chi^2$ for this scenario. How close is it to the critical value of $\chi^2 = 75.624$ obtained by using Statdisk and Minitab?
Based on the accompanying method of approximation, the critical value of $\chi^2$ for this scenario is. This estimate is of $\chi^2 = 75.624$ obtained by using Statdisk and Minitab.
(Round to two decimal places as needed.)
Approximation Information
For large numbers of degrees of freedom, we can approximate critical values of $\chi^2$ as follows:
$\chi^2 = (z + \sqrt{2k - 1})^2$
Here k is the number of degrees of freedom and z is the critical value(s) found from technology or a standard normal distribution table. In this scenario we have df=57, so a chi-square distribution table does not list an exact critical value. If we want to approximate a critical value of $\chi^2$ in the right-tailed hypothesis test with $\alpha = 0.05$ and a sample size of 58, we let k = 57 with z = 1.64 (or the more accurate