SECTION 10.2 Independence 537 Finding Expected Frequencies Find the expected frequency for each cell in the contingency table on the preceding page. Assume that the variables student’s living arrangement and family college experience are independent. SOLUTION The marginal frequencies are shown in the table below. Student’s living arrangement Family college experience With parents, rent free With parents, pay rent On campus Off campus, w/others Off campus, alone Total First in family 190 39 97 43 17 386 2nd generation 651 109 428 349 61 1598 Total 841 148 525 392 78 1984 To find each expected frequency, use the formula Expected frequency Er, c = 1Sum of row r2 # 1Sum of column c2 Sample size . The expected frequencies for the first row are E1, 1 = 386# 841 1984 ≈ 163.622 E1, 2 = 386# 148 1984 ≈ 28.794 E1, 3 = 386# 525 1984 ≈ 102.142 E1, 4 = 386# 392 1984 ≈ 76.266 E1, 5 = 386# 78 1984 ≈ 15.175 and the expected frequencies for the second row are E2, 1 = 1598# 841 1984 ≈ 677.378 E2, 2 = 1598# 148 1984 ≈ 119.206 E2, 3 = 1598# 525 1984 ≈ 422.858 E2, 4 = 1598# 392 1984 ≈ 315.734 E2, 5 = 1598# 78 1984 ≈ 62.825. TRY IT YOURSELF 1 The contingency table shows the results of a random sample of 1996 undergraduate students classified by their living arrangement and whether they borrowed money to pay for college. Find the expected frequency for each cell. Assume that the variables student’s living arrangement and borrowing status are independent. (Adapted from Sallie Mae/Ipsos) Student’s living arrangement Borrowing status With parents, rent free With parents, pay rent On campus Off campus, w/others Off campus, alone Borrowed 364 92 303 203 42 Did not borrow 484 58 236 186 28 Answer: Page A42 EXAMPLE 1 Study Tip In Example 1, after finding E1, 1 ≈ 163.622, you can find E2, 1 by subtracting 163.622 from the first column’s total, 841. So, E2, 1 ≈ 841 - 163.622 = 677.378. In general, you can find the expected value for the last cell in a column by subtracting the expected values for the other cells in that column from the column’s total. Similarly, you can do this for the last cell in a row using the row’s total.
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