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?

Step by step solution

01

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.

02

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.

03

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).

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