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Chapter 12: Problem 15

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. For a Chi-square test based on six campuses and three categories, the degrees of freedom will be 10. b. If there are seven campuses rather than six, the degrees of freedom will change to 12. c. If there are four categories for political philosophy rather than three, the degrees of freedom will change to 15.

Step by step solution

01

Understand the formula for degrees of freedom in Chi-square Test

The degrees of freedom in a Chi-Square distribution for testing homogeneity and independence in a contingency table is given by: \((R - 1) * (C - 1)\), where:R = number of rows, andC = number of columns. In this case, consider each campus as a row, and each political philosophy category as a column.
02

Calculate the degrees of freedom for six campuses, three categories

Substitute R = 6 and C = 3 into the formula to discover the degrees of freedom: \((6 - 1) * (3 - 1) = 5 * 2 = 10\). There are 10 degrees of freedom in this Chi-Square test.
03

Calculate the degrees of freedom if there are seven campuses

If there鈥檚 an additional campus, then R would be 7. Substituting R = 7 and C = 3 to the formula: \((7 - 1) * (3 - 1) = 6 * 2 = 12\). So, the degrees of freedom for this Chi-square test would be 12.
04

Calculate the degrees of freedom if there are four categories

If there鈥檚 an additional political philosophy category, then C would be 4. Substituting R = 6 and C = 4 into the formula: \((6 - 1) * (4 - 1) = 5 * 3 = 15\). Therefore, the degrees of freedom for this Chi-square test would be 15.

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Most popular questions from this chapter

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