Bluman A.G. Elementary Statistics- A Step By Step Approach

Section 11–2 Tests Using Contingency Tables 607I nteresting FactsYou’re never tooold—or too young—tobe your best. GeorgeForeman won theworld heavyweightboxing championshipat age 46. William Pittwas 24 when hebecame prime ministerof Great Britain.Benjamin Franklin wasa newspaper columnistat age 16 and a framerof the Constitutionwhen he was 81.A contingency table is designated as an R C (rows by columns) table. In this case,R 2 and C 3; hence, this table is a 2 3 contingency table. Each block in the tableis called a cell and is designated by its row and column position. For example, the cell witha frequency of 80 is designated as C 1,2 , or row 1, column 2. The cells are shown below.Column 1 Column 2 Column 3Row 1 C 1,1 C 1,2 C 1,3Row 2 C 2,1 C 2,2 C 2,3The degrees of freedom for any contingency table are (rows 1) times(columns 1); that is, d.f. (R 1)(C 1). In this case, (2 1)(3 1) (1)(2) 2.The reason for this formula for d.f. is that all the expected values except one are free tovary in each row and in each column.Using the previous table, you can compute the expected frequencies for each block(or cell), as shown next.1. Find the sum of each row and each column, and find the grand total, as shown.Prefer new Prefer old NoGroup procedure procedure preference TotalRow 1 sumNurses 100 80 20 200Row 2 sumDoctors 50 120 30 200Total 150 200 50 400Column 1 sum Column 2 sum Column 3 sum Grand total2. For each cell, multiply the corresponding row sum by the column sum and divideby the grand total, to get the expected value:row sum column sumExpected value grand totalFor example, for C 1,2 , the expected value, denoted by E 1,2 , is (refer to the previoustables)E 1,2 200200400For each cell, the expected values are computed as follows:E 1,1 200150400E 2,1 200150400 100 75 75E 1,2 200200400E 2,2 200200400 100 100E 1,3 20050400E 2,3 20050400 25 25The expected values can now be placed in the corresponding cells along with theobserved values, as shown.Prefer new Prefer oldGroup procedure procedure No preference TotalNurses 100 (75) 80 (100) 20 (25) 200Doctors 50 (75) 120 (100) 30 (25) 200Total 150 200 50 400The rationale for the computation of the expected frequencies for a contingency tableuses proportions. For C 1,1 a total of 150 out of 400 people prefer the new procedure. Andsince there are 200 nurses, you would expect, if the null hypothesis were true,(150400)(200), or 75, of the nurses to be in favor of the new procedure.11–17