Category percentage marked decrease in blood pressure

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Category Percentage Marked decrease in blood pressure 50% Moderate decrease in blood pressure 25% Sight decrease in blood pressure 10% Stationary or slight increase in blood pressure 15% A clinical trial is conducted with a random sample of 200 patients suffering from high blood pressure. All patients are required to be listed according to the same hypertensive categories of the NYHA Classification as those studied under the standard preparation. Use the sample data in the following table to test the hypothesis that the cell probabilities associated with the test preparation are identical to those for the standard. Use α = .05. Category Observed Cell Counts 1 120 2 60 3 10 4 10 SOLUTION This experiment possesses the characteristics of a multino- mial experiment, with n = 200 and k = 4 outcomes. Outcome 1: A person’s blood pressure will decrease markedly after treatment on the test drug. Outcome 2: A person’s blood pressure will decrease moderately after treat- ment on the test drug. Outcome 3: A person’s blood pressure will decrease slightly after treatment on the test drug. Outcome 4: A person’s blood pressure will remain stationary or increase slightly after treatment on the test drug. The null and alternative hypotheses are then H 0 : π 1 = . 50 , π 2 = . 25 , π 3 = . 10 , π 4 = . 15 12
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and H a : At least one of the cell probabilities is different from the hypothesized value. Before computing the test statistic, we must determine the expected cell numbers. These data are given in Table 10.3. Observed Cell Expected Cell Category Number , n i Number , E i 1 120 200(.50) = 100 2 60 200(.25) = 50 3 10 200(.10) =20 4 10 200(.15) = 30 Since all the expected cell numbers are large, we may calculate the chi-square statistic and compare it to a tabulated value of the chi-square distribution χ 2 = X i ( n i - E i ) 2 E i = (120 - 100) 2 100 + (60 - 50) 2 50 + (10 - 20) 2 20 + (10 - 30) 2 30 = 4 + 2 + 5 + 13 . 33 = 24 . 33 . For the probability of a Type I error set at α = . 05, we look up the value of the chi-square statistic for a = . 05 and df = k - 1 = 3. The critical value from Table 5 in the Appendix is 7.815. R.R. : Reject H 0 if χ 2 > 7 . 815 Conclusion: Since the computed value of χ 2 is greater than 7.815, we reject the null hypothesis and conclude that at least one of the cell probabilities dif- fers from that specified under H 0 . Practically, it appears that a much higher proportion of patients treated with the test preparation fall into the moderate and marked improvement categories. / £ ¡ ¢ EXAMPLE 10.7 Gregor Mendel (1822-1884) was an Austrian monk whose genetic theory is one of the greatest scientific discovery of all time. In his famous experiment with garden peas, he proposed a genetic model that would explain the occurrence of hereditary characteristics. In particular, he studied how the shape (smooth or wrinkled) and color (yellow or green) of pea seeds are transmitted through 13
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generations. According to his model, the second generation of peas from a certain ancestry should have the following distribution: wrinkled - green wrinkled - yellow smooth - green smooth - yellow 1/16 3/16 3/16 9/16 The outcome of his experiment was as follows: wrinkled - green wrinkled - yellow smooth -
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