Researchers propose a new treatment for a disease. Instead of recruiting

patients in the experiment to be a control group, researchers decide to use historical controls as below: Study ID #Patients #Response 1 65 54 2 39 20 3 133 59 4 37 21 5 150 82 (a) [10 points] Suppose there is no study-to-study variation, then the true response rates for studies should be equal, i.e., =2 = ... = 5 = . Please estimate 7 and also calculate the standard error of the estimate. (b) [15 points] Suppose there is study-to-study variation, then the true response rates 1, T2, ..., 5 for studies are iid with (4,0%). An estimate of is denoted by . Please calculate and its corresponding standard error. (c) [15 points] Suppose there is study-to-study variation, then the true response rates T1, T2, ..., 5 for studies are iid with (x, 02). Please estimate .