11-2 Contingency Tables 591 The word contingent has a few different meanings, one of which refers to a dependence on some other factor. We use the term contingency table because we test for independence between the row and column variables. We first define a test of independence and we provide key elements of the test in the Key Elements box that follows. DEFINITION In a test of independence, we test the null hypothesis that in a contingency table, the row and column variables are independent. (That is, there is no dependency between the row variable and the column variable.) Contingency Table Objective Conduct a hypothesis test of independence between the row variable and column variable in a contingency table. Notation O represents the observed frequency in a cell of a contingency table. E represents the expected frequency in a cell, found by assuming that the row and column variables are independent. r represents the number of rows in a contingency table (not including labels or row totals). c represents the number of columns in a contingency table (not including labels or column totals). Requirements 1. The sample data are randomly selected. 2. The sample data are represented as frequency counts in a two-way table. 3. For every cell in the contingency table, the expected frequency E is at least 5. (There is no requirement that every observed frequency must be at least 5.) KEY ELEMENTS Null and Alternative Hypotheses The null and alternative hypotheses are as follows: H0: The row and column variables are independent. H1: The row and column variables are dependent. Test Statistic for a Test of Independence x2 = a1O - E22 E where O is the observed frequency in a cell and E is the expected frequency in a cell that is found by evaluating E = 1row total2 1column total2 1grand total2 P-values P-values are typically provided by technology, or a range of P-values can be found from Table A-4. Critical values: 1. The critical values are found in Table A-4 using Degrees of freedom = 1r − 12 1c − 12 where r is the number of rows of data and c is the number of columns of data (not including labels). 2. Tests of independence with a contingency table are always right-tailed.
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