Improving SAT scores. Chance (Winter 2001) published a study of students who paid a private coach to help them improve their Scholastic Assessment Test (SAT) scores. 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). Based on the interval, 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, part e, was statistically significant at α = .05. Practically interpret this result. (G). The alternative model, part d, resulted in R2 a = .79, βˆ2 = 14, and s ˆβ2 value of R2 a. (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, parts d and j. How would you perform this test?
Improving SAT scores. Chance (Winter 2001) published a study of students who paid a private coach to help them improve their Scholastic Assessment Test (SAT) scores. 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). Based on the interval, 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, part e, was statistically significant at α = .05. Practically interpret this result.
(G). The alternative model, part d, resulted in R2 a = .79, βˆ2 = 14, and s ˆβ2 value of R2 a.
(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, parts d and j. How would you perform this test?
Solution :-
(A) The adjusted R2 value of 0.76 means that the model with x1 and x2 as predictors explains 76% of the total variation in the SAT-Math scores, after adjusting for the degrees of freedom.
(B) The 95% confidence interval for β2 is calculated as 19 ± 1.96(3), which gives a range of [12.2, 25.8]. This means that we are 95% confident that the true value of β2 lies in this range.
(C) The interval in part (B) indicates that the effect of coaching on SAT-Math scores is statistically significant, as the interval does not contain zero. This suggests that coaching has a positive effect on SAT-Math scores, as the estimate of β2 is positive.
(D) The hypothesized equation for E(y) for the alternative model would be E(y) = β0 + β1x1 + β2x2 + β3x3 + β4x4 + β5x5 + β6x6 + β7x7 + β8x8 + β9x9 + β10x10, where x3, x4, and x5 are dummy variables for student ethnicity, x6 is a socioeconomic status index variable, x7 and x8 are variables that measure high school performance, x9 is the number of math courses taken in high school, and x10 is the overall GPA for math courses.
(E) The null hypothesis for a nested model F-test comparing the initial and alternative models is that the alternative model does not provide a better fit than the initial model, i.e., the additional variables in the alternative model do not improve the fit of the model.
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