You must provide a complete explanation of how you got your answers. For questions involving data analysis, you must attach an output log of how you estimated your parameter values.

2. Consider the simple hypothetical example in Table 1. This example involves patients and two possible medical procedures labeled 0 (Drug) and 1 (Surgery). Table 1 displays each patient's potential outcomes in terms of years of post-treatment survival under each treatment. PatientYl Y\" 1 8 l 2 5 7 3 5 l 4 6 8 5 4 2 6 10 2 7 l 10 8 4 6 9 3 7 10 9 5 a. What is the average treatment effect for surgery compared to drug? Which type of intervention is more effective on average? b. Suppose that there is some \"perfect doctor\" who through expertise or magic knows each patient's potential outcomes and then chooses the optimal treatment for each patient. If she assigns each patient to the treatment more benecial for that patient, which patients will receive surgery and which will receive the drug treatment? Calculate the average treatment effect both for the group who received surgery and the group who received drug treatment. c. Assume that the \"perfect doctor\" has assigned each patient to his or her own optimal treatment. Calculate the naive average treatment effect for surgery using that observable information? How does this differ from the ATE? d. Decompose the naive average treatment effect into the baseline and differential effects biases. In your own words, what is the cause of the difference between the \"real\" average treatment effect of surgery and the naive estimate you obtained? e. What is the \"andamental problem of causal inference\" in this context? f. Take a coin and ip it ten times. If heads on the rst ip, unit 1 is assigned to drug and if tails it is assigned to surgery. If heads on the second ip, unit 2 is assigned to drug and if tails it is assigned to surgery. And so on. Calculate the estimated average treatment effect and compare it with your answers