Question

Finding Critical Values of${\chi }^2$For large numbers of degrees of freedom, we can approximate critical values of${\chi }^2$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${\chi }^2$in the right-tailed hypothesis test with $\alpha$= 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${\chi }^2$for Exercise 12. How close is it to the critical value of${\chi }^2$= 82.292 obtained by using Statdisk and Minitab?

Short answer

The estimated critical value is equal to 81.54. The value obtained is quite close to the value obtained using technology (82.292).

Step-by-step solution

Given information

A sample of the number of words spoken in a day is considered.

The sample size is equal to 56. The value of the degrees of freedom is equal to 55.

The value of the z-score is equal to 2.33. The actual critical value of is equal to 82.292.

Compute the approximate critical value 

The approximate critical value has the following formula:

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

The values are given as follows.

• k is equal to 55.
• The z-score is equal to 2.33.

Substitute the above values in the formula to obtain the critical value, as shown.

$$\begin{gathered} {\chi }^2=\frac{1}{2}{\left(2.33+\sqrt{2\left(55\right)-1}\right)}^2 \\ \approx 81.54 \end{gathered}$$

Thus, the critical value is equal to 81.54.

Comparison

The critical value of ${\chi }^2$ obtained using the formula (81.54) is quite close to the critical value of ${\chi }^2$ obtained using technology (82.292).