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Chapter 12: Problem 14

A particular state university system has six campuses. On each campus, a random sample of students will be selected, and each student will be categorized with respect to political philosophy as liberal, moderate, or conservative. The null hypothesis of interest is that the proportion of students falling in these three categories is the same at all six campuses. a. On how many degrees of freedom will the resulting \(X^{2}\) test be based? b. How does your answer in Part (a) change if there are seven campuses rather than six? c. How does your answer in Part (a) change if there are four rather than three categories for political philosophy?

Short Answer

Expert verified
a. The degrees of freedom for the resulting \(X^{2}\) test will be 10.\nb. If there were seven campuses instead of six, the degrees of freedom would increase to 12.\nc. If there are four categories for political philosophy instead of three, the degrees of freedom will increase to 15.

Step by step solution

01

Analyzing the given data

The total number of categories of political philosophy is three. The total number of campuses is mentioned as six, hence the number of groups is six. The formula to calculate degrees of freedom in a chi-square test is (C-1)*(R-1), where C is the number of columns and R is the number of rows. In this instance, the philosophy categories can be considered as columns and campuses as rows.
02

Calculating Degrees of Freedom for Part a

Using the formula (C-1)*(R-1), substitute C=3 (liberal, moderate and conservative) and R=6 (number of campuses). Therefore df = (3-1) * (6-1)= 2*5= 10. The degrees of freedom for resulting \(X^{2}\) test based on six campuses and three political philosophies is 10.
03

Adapting the calculation for Part b

For part b, the number of campuses increased to seven from six. Substituting C=3 (liberal, moderate and conservative) and R=7 (number of campuses) into the formula, we get df = (3-1) * (7-1) = 2*6 = 12. So, chances are the degrees of freedom will increase to 12 if there are seven campuses.
04

Adapting the calculation for Part c

For part c, there are now four categories for political philosophy instead of three. Substitute C=4 and R=6 (number of campuses) into the formula. Hence, df = (4-1) * (6-1) = 3*5 = 15. Thus, the degrees of freedom now changed to 15 if there are four categories for political philosophy.

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