A medical researcher wants to test the efficacy of a drug for a population that has a certain health condition. With exercise only, 35% of both types get better. The drug does not help type A people but increases the percentage of type B people who recover to 75% (see table below). The researcher does not know this information and conducts an experiment to attempt to uncover it. Assume everyone who has this particular condition and is participating in the research exercises - 50% of participants are Type A and the rest are Type B. Round all final results to two decimal digits. Type B Exercise Only 0.35 0.35 Exercise + Drug 0.35 0.75 A. (5 points) Given the information in the table, what fraction of people with the health condition is expected to recover if no one receives the drug? What fraction is expected to recover if everyone with the condition receives the drug? What is the percentage increase in the probability of recovery due to the drug for the population at large? B. (10 points) The researcher has a sample of 2,000 people with the condition; 1000 are randomly assigned to the treatment group (those who get the drug and exercise) and 1000 to the control group (those who get only exercise). Based on the actual recovery probabilities in the table, what fraction of each group in the sample is expected to get better? Using these estimates, what is the expected percentage increase in the probability of recovery due to the drug? [Hint: If half the people in the treatment group have a 0.75 probability of recovery, and half have a 0.35 probability of recovery, what is the expected probability of recovery for a person randomly drawn from the treatment group? Then ask the same question of someone randomly drawn from the control group and compare the two probabilities.] C. (10 points) Suppose that instead of random assignment, type B people tend to self-select into the treatment group, resulting in 75% of the treatment group being type B. The control group ends up being 25% type B. The researcher is unaware of this self-selection and proceeds as if group assignment had been random. What is the expected estimate of the increase in the probability of recovery due to the drug, based on this sample, when the researcher ignores self-selection? How does the failure to correct for self-selection bias the estimate of the efficacy of the drug? [Hint: Under the assumptions, 70% of people in the treatment group have a 0.7 probability of recovery, and 30% have a 0.4 probability of recovery. What is the probability that someone randomly drawn from the treatment group recovers? Then ask the same question of someone randomly drawn from the control group and compare the recovery probabilities of the two groups.]