(b) (2 points) Suppose you are interested in conducting a study where you compare the average MATH 3B grades (Yi) of students enrolling in a special math tutoring pro- gram (treatment) versus those without the tutoring (control). However, you want to ensure that your study's design eliminates selection bias. How can you eliminate this issue when sorting your student participants within the context of this experiment? What can you do to maximize the effectiveness of your solution? Please make sure your response is specific and concise. Page 3

2. Selection Bias. Let Y, be some outcome variable for an individual i and D; be a binary variable that takes on a value of "1" if the individual receives some treatment. Recall the following potential outcomes framework: Y - U.(0) , if D, = 0 U.(1) , if Di = 1 where y,(1) is the observed outcome for those that receive the treatment while y;(0) is the outcome for those that did not receive it. Thus, Y, = D,y.(1) + (1 - D;)y.(0). In this framework, we estimate E(Y,[D, = 1) - E(Y,|D, = 0). (a) (6 points) As you've learned in this class, researchers often deal with the issue of selection bias, which could result from the faulty selection of participants for an experimental study. Consequently, E(Y,| D, = 1) - E(Y,|D, = 0) can be decomposed into two components: the average treatment effect on the treated (ATT) and selection bias. This decomposition is provided below: E(Y.D. = 1) - E(YD, = 0) - Ey.(1) - y.(0)|D, - 1] ATT + Ely.(0) |D, = 1] - Ev.(0)|D. = oj selection bias Derive this decomposition. Make sure to provide a justification for each step of your derivation. Hint: Here, you can invoke the linearity of conditional expectations, E(X + YIZ) - E(X Z) + E(Y|Z) where X, Y, and Z are random variables