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 \({\chi ^2}\) 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.

Substitute the above values in the formula to obtain the critical value, as shown.
\(\begin{array}{c}{\chi ^2} = 55{\left( {1 - \frac{2}{{9 \times 55}} + 2.33 \times \sqrt {\frac{2}{{9 \times 55}}} } \right)^3}\\ \approx 82.360\end{array}\).

Thus, the critical value is equal to 82.360.

Comparison

The critical value of\({\chi ^2}\)obtained using the formula (82.360) isapproximately equalto the critical value of \({\chi ^2}\) obtained using technology (82.292).