STAT 400 Lecture AL1 Examples for 9.2 Spring 2015 Dalpiaz The χ 2 Test of Homogeneity ( one margin fixed ) Independent random samples from r populations. Each sample classified in c response categories. H 0 : In each response category, the probabilities are equal for all r populations. The χ 2 Test of Independence ( neither margin fixed ) A random sample of size n is simultaneously classified with respect to two characteristics, one has r categories and the other c categories. H 0 : The two classifications are independent; that is, each cell probability is the product of the row and column marginal probabilities. Test Statistic: ( )∑−=cellsEEOQ2 ×==totalgrandtotalcolumntotalrowEfrequencycellobservedO Rejection Region: Reject H 0 if Q ≥ 2αχ, d.f. = ( No. of rows – 1 ) × ( No. of columns – 1 ) = ( r – 1 ) × ( c – 1 )1. We wish to test whether the proportions of individuals with each of the four blood types are the same in two neighboring towns, Town X and Town Y. A random sample of 300individuals from Town X and 200 individuals from Town Y produced the following observed frequencies: Blood Type O A B AB Town X 120 85 60 35 300 Town Y 100 45 30 25 200 220 130 90 60 500 Use α = 0.05 to test H 0 : p XO = p YO , p XA = p YA , p XB = p YB , p XAB = p YAB .2. In a random sample of 500 voters, each individual was asked whether he or she thought inflation of unemployment was a more serious problem. The individuals were also classified by party affiliation. The results were as follows: Party Unemployment Inflation Democrat 150 70 Republican 100 80 Other 60 40 Use a 5% level of significance and test whether political party affiliation and perceived problem are independent.3. In a comparative study of two new drugs, A and B, 120 patients were treated with drug A and 150 patients with drug B, and the following results were obtained. Drug A Drug B Cured 78 111 Not cured 42 39 Total 120 150 We wish to test whether drug A and drug B have the same cure rate. H 0 : p AC = p BC , p AN = p BN . Recall: 12078Y111 ˆ==np = 0.65. 150111Y222 ˆ==np = 0.74. 27018915012011178YY2121 ˆ=++=++=nnp = 0.70. Test Statistic: Z = +−⋅⋅1501120130.070.0 74.065.0 ≈ –
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