Question

7. The following data summarizes the incidence of Coronary Heart Disease (CHD) for people who smoked regularly and for pe ople who did not smoke regularly : Smoked Did not Smoke CHD No CHD 87 4913 84 2916 (a) State the null and alternative hypothesis for testing if CHD status is independent of smoking status (b) Calculate theLikelihood-Ratio test-statistic and its corresponding p-value (c) State your conclusion in terms of the problem if 0.05 (d) Describe what "more extreme"would mean in terms of this null. What would it mean for our data to be "different" than the null (e) Can we say with statistical evidence that smokers have a higher chance of having CHD based on this test? Explain

Answer 1

a) From the given information, the null and altenate hypotheses can be stated as follows: Null hypothesis: the CHD status and the smoking status are not independent. Alternate hypothesis: the CHD status and the smoking status are independent. b) Calculate the likelihood ratio test statistic and its corresponding P-value as follows: G likelihood -ratio test: O. 2ΣQ E, G = The given contingency table is, Total Observed values CHD NO CHD 171 Smoked 84 87 2916 Did not smoked 4913 7829 Total 3000 5000 8000 Calculate the expected values as follows: (Row total)(Column total) Expected frequencies Overall total value 171x 3000 = 64.125 E1 8000 7829 x 5000 En 4893.1 8000 The resulted expected values table is as shown below: NO CHD Total Expected values CHD Smoked 64.125 106.88 171 Did not smoked 2935.9 4893.1 7829 Total 3000 8000 5000 Calculate the test statistic as follows: Observed Expected values 0xm Ox In In values 0.270 84 64.125 1.310 22.679 -0.206 87 106.88 0.814 -17.904 -0.007 2916 2935.9 0.993 -19.832 0.004 19.940 4913 4893.1 1.004 Total O, = 4.882 Σο G=2Σ0 n = 2 x 4.882 = 9.764 P = CHIDIST (G, df) df = (r –1)(c -1) = (2–1)(2 – 1) P = CHIDIST (9.764,1) = 0.0271 c) From the above calculations, the P-value of the test is 0.0271. Which is less than the level of significance a = 0.05 i.e. (0.0271<0.05).Thus, the researcher can reject the null hypotheses. The CHD status and the smoking status are independent. d) There is 2.71% of possibility of the sample regions being more extreme for being the CHD status and the smoking status are dependent. it means for our data to be "different" than the null is the CHD status and the smoking status are independent.