3. Consider the regression model (a) Use either calculus or first differences to show the following. ii. /2-(1/X2) (A + 2As log(x2) +3.x,). (Hint: Use the fact that log(X2 + (b) Suppose that a researcher collects data on test scores of schools in Massachusetts and obtains the regression results in table 3b below. The dependent variable is TestScore, the average test score recorded in a given school; ST R is the student teacher ratio in that school; and log(Income) is the log of the average income recorded in the county where the school is located. (Intercept) 619.39 STR -13.65 106.83 log(Income24.42 4.48 Estimate Std. Error t value Pr(> t) 0.00 0.00 0.03 0.00 0.00 93.48 6.63 2.77-4.93 9.87 2.14 6.57-3.72 0.94 4.79 logIncome STR log(Income) i. What happens in this model when STR increases by 1 unit? Why is the coefficient corresponding to the interaction term STR x log(Income) positive? Explain. 11. Suppose that log(Incorne) = 2.5. What is the expected change of an increase in STR of 3 students per teacher? Construct a 95% confidence interval for the change in TestScore, knowing that Cov,2.5735 3. Consider the regression model (a) Use either calculus or first differences to show the following. ii. /2-(1/X2) (A + 2As log(x2) +3.x,). (Hint: Use the fact that log(X2 + (b) Suppose that a researcher collects data on test scores of schools in Massachusetts and obtains the regression results in table 3b below. The dependent variable is TestScore, the average test score recorded in a given school; ST R is the student teacher ratio in that school; and log(Income) is the log of the average income recorded in the county where the school is located. (Intercept) 619.39 STR -13.65 106.83 log(Income24.42 4.48 Estimate Std. Error t value Pr(> t) 0.00 0.00 0.03 0.00 0.00 93.48 6.63 2.77-4.93 9.87 2.14 6.57-3.72 0.94 4.79 logIncome STR log(Income) i. What happens in this model when STR increases by 1 unit? Why is the coefficient corresponding to the interaction term STR x log(Income) positive? Explain. 11. Suppose that log(Incorne) = 2.5. What is the expected change of an increase in STR of 3 students per teacher? Construct a 95% confidence interval for the change in TestScore, knowing that Cov,2.5735