In the STAR experiment (Section 7.5.3), children were randomly assigned within schools into three types of classes: small classes with 13-17 students, regular-sized classes with 22-25 students, and regular-sized classes with a full-time teacher aide to assist the teacher. Student scores on achievement tests were recorded as well as some information about the students, teachers, and schools. Data for the kindergarten classes is contained in the data file star5_small2.

a. Calculate the average of MATHSCORE for (i) students in regular-sized classrooms with full-time teachers but no aide; (ii) students in regular-sized classrooms with full-time teachers and an aide; and (iii) students in small classrooms. What do you observe about test scores in these three types of learning environments?

b. Estimate the regression model MATHSCORE MA \(_{i}+\beta_{2} \operatorname{SMALL}_{i}+\beta_{3} A I D E_{i}+e_{i}\), where AIDE is an indicator variable equaling 1 for classes taught by a teacher and an aide, and 0 otherwise. What is the relation of the estimated coefficients from this regression to the sample means in part (a)? Test the statistical significance of \(\beta_{3}\) at the \(5 \%\) level.

c. To the regression in (b) add the additional explanatory variable TCHEXPER. Is this variable statistically significant? Does its addition to the model affect the estimates of \(\beta_{2}\) and \(\beta_{3}\) ? Construct a 95\% interval estimate of expected math score for a student in a small class with a teacher having 10 years of experience. Construct a \(95 \%\) interval estimate of expected math score for a student in a class with an aide and having a teacher with 10 years of experience. Calculate the least squares residuals from this model, calling them EHAT. This variable will be used in the next part.

d. To the regression in (c), add the additional indicator variable FREELUNCH. Students from lower income households receive a free lunch at school. Is this variable statistically significant? Does its addition to the model affect the estimates of \(\beta_{2}\) and \(\beta_{3}\) ? What explains the sign of FREELUNCH? Calculate the sample average of \(E H A T\), from part (c), for students receiving a free lunch, and for students who do not receive a free lunch. Are the residual averages consistent with the regression that includes FREELUNCH?

e. To the model in (d), add interaction variables between FREELUNCH and SMALL, AIDE and TCHEXPER. Are any of these individually significant? Test the joint significance of these three interaction variables at the \(5 \%\) level. What do you conclude?

f. Carry out a Chow test for the equivalence of the regression MATHSCORE MA \(_{i}+\beta_{2} \operatorname{SMALL}_{i}+\) \(\beta_{3} A I D E_{i}+\beta_{4}\) TCHEXPER \(+e_{i}\) for students who receive a free lunch and those who do not receive a free lunch. How does this test result compare to the test result in part (e)?