Show that the model can always be rewritten with the same slope, but a new intercept and error, where the new error has a zero expected value.
Suppose you are interested in estimating the effect of hours spent in an SAT preparation course for seniors in a particular year.
1. Suppose you are given a grant to run a controlled experiment. Explain how you would structure the experiment in order to estimate the causal effect of hours on SAT.
2. Consider the more realistic case where students choose how much time to spend in a preparation course, and you can only randomly sample SAT and hours from the population. Write the population model as SAT = β0 + β1 * hours + u, where, as usual in a model with an intercept, we can assume E(u) = 0. List at least two factors contained in u. Are these likely to have a positive or negative correlation with hours?
3. In the equation from part 2, what should be the sign of β1 if the preparation course is effective?
4. In the equation from part 2, what is the interpretation of β0?
3. The following table contains the ACT scores and the GPA (grade point average) for eight college students. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal.
Student GPA ACT
1 2.7 21
2 3.4 24
3 3.0 26
4 3.5 27
5 3.6 29
6 3.0 25
7 3.7 30
