Question

Finding Critical Values of${\chi }^2$Repeat Exercise 19 using this approximation (with k and z as described in Exercise 19):

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

Short answer

The estimated critical value is equal to 82.360. The value obtained is approximately equal 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:

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

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.

$$\begin{gathered} {\chi }^2=55{\left(1-\frac{2}{9\times 55}+2.33\times \sqrt{\frac{2}{9\times 55}}\right)}^3 \\ \approx 82.360 \end{gathered}$$

Thus, the critical value is equal to 82.360.

Comparison

The critical value of ${\chi }^2$obtained using the formula (82.360) is approximately equal to the critical value of ${\chi }^2$ obtained using technology (82.292).