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., 1 = 2 = ... = 5 = . Please estimate 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, 2, ..., 5 for studies are iid with (, 2). 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 1, 2, ..., 5 for studies are iid with (, 2). Please estimate 2