Final_Review_1_Solutions

Each patient entering the program was designated

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Unformatted text preview: aning that the probability of a negative test result when a person does not actually have the predisposition is 0.98. What is the probability that a randomly selected person who is identiﬁed as having the predisposition by the test actually has the predisposition? P (pre and positive) P (positive) 0.0297 = 0.0297 + 0.0194 = 0.6049 P (pre|positive) = Result Predisposition? 0.99 0.99*0.03 = 0.0297 0.01 0.01*0.03 = 0.0003 positive yes 0.03 0.97 no negative positive negative 0.02 0.98 0.02*0.97 = 0.0194 0.98*0.97 = 0.9506 13 18. If you roll a pair of fair dice, what is the probability of (a) getting a sum of 1? P(sum of 1) = 0. Since there are two dice being rolled, the minimum possible sum is 2. (b) getting a sum of 5? P(sum of 5) = P(1,4) + P(4,1) + P(2,3) + P(3,2) = (c) getting a sum of 12? P(sum of 12) = P(6,6) = ￿1 6 ∗ 1 6 ￿ = ￿1 6 ∗ 1 6 ￿ *4= 4 36 = 0.11. 1 36 . 19. The Stanford University Heart Transplant Study was conducted to determine whether an experimental heart transplant program increased lifespan. Each patient entering the program was designated oﬃcially a heart transplant candidate, meaning that he was gravely ill and would most likely beneﬁt from a new heart. Some patients got a transplant and some did not. The variable group indicates what group the patients were in; treatment group got a transplant and control group did not. Another variable in the study, outcome, indicates whether or not the patient was alive at the end of the study. introduces the Stanford Heart Transplant Study. Of the 34 patients in the control group, 4 were alive at the end of the study. Of the 69 patients in the treatment group, 24 were alive. The contingency table below summarizes these results. Outcome Alive Dead Total Group Control Treatment 4 24 30 45 34 69 Total 28 75 103 (a) What proportion of all patients died? Proportion of all patients who died: 75 103 = 0.73 (b) If outcome and treatment were independent (i.e. there was no diﬀerence between the success rates of two the groups), about how many deaths would we have expected in the treatment group? Expected number of deaths in the treatment group if outcome and treatment were independent is calculated as the number of patients in that group multiplied by the overall death rate in the study: 69 ∗ 0.73 ≈ 50. (c) Using a randomization technique, a researcher investigated the relationship between outcome and treatment in this study. In order to simulate from the independence model, she wrote whether or not each patient survived on cards, shuﬄed all the cards together, then dealt them into two groups of size 69 and 34. She repeated this simulation 250 times (using the help of a statistical software) and each time recorded the number of patients who died in the treatment group. Below is a histogram of these counts. i. What are the claims being tested? H0 : The variables group and outcome are independent. They have no relationship, and the diﬀerence in survival rates between the control and treatment groups was due to chance. In other words, heart transplant is not eﬀective. HA : The variables group and outcome are not independent. The diﬀerence in survival rates between the control and treatment groups was not due to chance and the heart transplant is eﬀective. 14 ii. Would more deaths or fewer deaths in the treatment group than the number calculated in part (b) provide support for the alternative hypothesis? A fewer number of deaths than expected under the assumption of independence would provide support for the alternative hypothesis as this would suggest that the treatment group has a higher survival rate. iii. What do the simulation results suggest about the eﬀectiveness of the transplant program? Use the p-value to support your answer. 20 10 0 45 50 55 trmtDead Under the independence model, only 2 out of 250 times (a probability of 0.008) did we get 45 or fewer deaths in the treatment group. Since this is such a low probability, we can reject the claim of independence in favor of the alternate model. There is convincing evidence to suggest that the transplant program is eﬀective. 15...
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