Independence 10.2 536 CHAPTER 10 Chi-Square Tests and the F-Distribution What You Should Learn How to use a contingency table to find expected frequencies How to use a chi-square distribution to test whether two variables are independent ContingencyTables The Chi-Square IndependenceTest Contingency Tables Recall from Section 3.2 that two events are independent when the occurrence of one event does not affect the probability of the occurrence of the other event. For instance, the outcomes of a roll of a die and a toss of a coin are independent. But, suppose a medical researcher wants to determine whether there is a relationship between caffeine consumption and heart attack risk. Are these variables independent or are they dependent? In this section, you will learn how to use the chi-square test for independence to answer such a question. Performing this test involves using sample data that are organized in a contingency table. Anr : c contingency table shows the observed frequencies for two variables. The observed frequencies are arranged in r rows and c columns. The intersection of a row and a column is called a cell. DEFINITION A 2 * 5 contingency table is shown below. It has two rows and five columns and shows the results of a random sample of 1984 undergraduate students classified by two variables, student’s living arrangement and family college experience. From the table, you can see that of the students who live on campus, 97 are the first in their family and 428 are the second generation in their family to experience college. Student’s living arrangement Family college experience With parents, rent free With parents, pay rent On campus Off campus, w/others Off campus, alone First in family 190 39 97 43 17 2nd generation 651 109 428 349 61 (Adapted from Sallie Mae/Ipsos) Assuming two variables are independent, you can use a contingency table to find the expected frequency for each cell, as shown in the next definition. The expected frequency for a cell Er, c in a contingency table is Expected frequencyEr, c = 1Sum of rowr2 # 1Sum of columnc2 Sample size . Finding the Expected Frequency for Contingency Table Cells When you find the sum of each row and column in a contingency table, you are calculating the marginal frequencies. A marginal frequency is the frequency that an entire category of one of the variables occurs. For instance, in the table above, the marginal frequency for students who live with their parents and pay rent is 39 + 109 = 148. The observed frequencies in the interior of a contingency table are called joint frequencies. Study Tip In a contingency table, the notation Er, c represents the expected frequency for the cell in row r, column c. For instance, in the table above, E1, 4 represents the expected frequency for the cell in row 1, column 4. Study Tip Note that “2 * 5” is read as “two-by-five.”
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