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

Thus, the critical value is equal to 81.54.

03

Comparison

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

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!