Refer to the Chance (Winter 2001) study of students who paid a private tutor (or coach) to help them improve their SAT scores, presented in Exercise 2.197 (p. 108). Multiple regression was used to estimate the effect of coaching on SAT-Mathematics scores. Data on 3,492 students (573 of whom were coached) were used to fit the model E(y) = β0 + β1x1 + β2x2, where y = SAT-Math score, x1 = score on PSAT, and x2 = (1 if student was coached, 0 if not).
a. The fitted model had an adjusted R2 value of .76. Interpret this result.
b. The estimate of β2 in the model was 19, with a standard error of 3. Use this information to form a 95% confidence interval for β2. Interpret the interval.
c. On the basis of the interval you found in part b, what can you say about the effect of coaching on SAT-Math scores?
d. As an alternative model, the researcher added several "control" variables, including dummy variables for student ethnicity (x3, x4, and x5), a socioeconomic status index variable (x6), two variables that measured high school performance (x7 and x8), the number of math courses taken in high school (x9), and the overall GPA for the math courses (x10). Write the hypothesized equation for E(y) for the alternative model.
e. Give the null hypothesis for a nested-model F-test comparing the initial and alternative models.
f. The nested-model F-test from part e was statistically significant at α = .05. Interpret this result practically.
g. The alternative model from part d resulted in R2α = .79, β̂2 = 14, and sβ̂2 = 3. Interpret the value of R2α.
h. Refer to part g. Find and interpret a 95% confidence interval for β2.
i. The researcher concluded that "the estimated effect of SAT coaching decreases from the baseline model when control variables are added to the model." Do you agree? Justify your answer.
j. As a modification to the model of part d, the researcher added all possible interactions between the coaching variable (x2) and the other independent variables in the model. Write the equation for E(y) for this modified model.
k. Give the null hypothesis for comparing the models from parts d and j. How would you perform this test?