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Improving SAT scores. Refer to the Chance (Winter 2001) study of students who paid a private tutor (or coach) to help them improve their Standardized Assessment Test (SAT) scores. Recall that multiple regression was used to estimate the effect of coaching on SAT–Mathematics scores, where y = SAT–Math score x1 = score on PSAT x2 = {1 if student was coached, 0 if not}

(a) Write a complete second-order model for E(y) as a function of x1 and x2.

(b) Give the equation of the curve relating E(y) to x1 for noncoached students. Identify the y-intercept, shift parameter, and rate of curvature in the equation.

(c) Repeat part b for students who have been coached on the SAT.

(d) How would you test to determine if coaching has an effect on SAT–Math scores?