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) Suppose there is no study-to-study variation, then the true response rates for studies should be equal, i.e., \pi 1 = \pi 2 = ... = \pi 5 = \pi . Please estimate \pi and also calculate the standard error of the estimate. (b) Suppose there is study-to-study variation, then the true response rates \pi 1,\pi 2,...,\pi 5 for studies are iid with (\pi ,\sigma 2 \pi ). An estimate of \pi is denoted by \pi . Please calculate \pi and its corresponding standard error. (c) Suppose there is study-to-study variation, then the true response rates \pi 1,\pi 2,...,\pi 5 for studies are iid with (\pi ,\sigma 2 \pi ). Please estimate \sigma 2 \pi