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You are interested in the causal impact of a child development program on year 11 math test scores. Children start year 11 when they are

You are interested in the causal impact of a child development program on year 11 math test scores. Children start year 11 when they are 11 years old. The child development program runs for 10 weeks over in the summer between year 10 and year 11 and is a 1-day class per week of math exercises. These additional classes are designed to help consolidate the mathematical content the students were taught in year 10. Suppose you know that children who performed worse in year 10 maths tests were more likely to go into the child development program. Let S represent the test score of a student in the year 11, X represent the test score of the student in year 10, and P=1 if a student is in the child development program and P=0 if not. Further, define the following potential outcomes: i) S_1 denotes the year 11 test score if P=1, and ii) S_0 denotes the year 11 test score if P=0. Answer the following questions: a) Using the potential outcomes framework, write out an expression for the expected observed effect of the child development program on year 11 test scores. In no more than two sentences, explain the

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