We introduced Tennessee's Project STAR (Student/Teacher Achievement Ratio) in Exercise 2.22. The data file is star5_small. [The data file star5 contains more observations and more variables.] Three types of classes were considered: small classes \([S M A L L=1]\), regular-sized classes with a teacher aide \([A I D E=1]\), and regular-sized classes \([\) REGULAR \(=1]\).

a. Compute the sample mean and standard deviation for student math scores, MATHSCORE, in small classes. Compute the sample mean and standard deviation for student math scores, MATHSCORE, in regular classes, with no teacher aide. Which type of class had the higher average score? What is the difference in sample average scores for small classes versus regular-sized classes? Which type of class had the higher score standard deviation?

b. Consider students only in small classes or regular-sized classes without a teacher aide. Estimate the regression model MATHSCORE \(=\beta_{1}+\beta_{2} S M A L L+e\). How do the estimates of the regression parameters relate to the sample average scores calculated in part (a)?

c. Using the model from part (b), construct a \(95 \%\) interval estimate of the expected MATHSCORE for a student in a regular-sized class and a student in a small class. Are the intervals fairly narrow or not? Do the intervals overlap?

d. Test the null hypothesis that the expected mathscore is no different in the two types of classes versus the alternative that expected MATHSCORE is higher for students in small classes using the \(5 \%\) level of significance. State these hypotheses in terms of the model parameters, clearly state the test statistic you use, and the test rejection region. Calculate the \(p\)-value for the test. What is your conclusion?

e. Test the null hypothesis that the expected MATHSCORE is 15 points higher for students in small classes versus the alternative that it is not 15 points higher using the \(10 \%\) level of significance. State these hypotheses in terms of the model parameters, clearly state the test statistic you use, and the test rejection region. Calculate the \(p\)-value for the test. What is your conclusion?

Data From Exercise 2.22:-

A longitudinal experiment was conducted in Tennessee beginning in 1985 and ending in 1989. A single cohort of students was followed from kindergarten through third grade. In the experiment 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 are contained in the data file star5_small.

a. Using children who are in either a regular-sized class or a small class, estimate the regression model explaining students' combined aptitude scores as a function of class size, TOTALSCORE \(E_{i}=\) \(\beta_{1}+\beta_{2} S M A L L_{i}+e_{i}\). Interpret the estimates. Based on this regression result, what do you conclude about the effect of class size on learning?

b. Repeat part (a) using dependent variables READSCORE and MATHSCORE. Do you observe any differences?

c. Using children who are in either a regular-sized class or a regular-sized class with a teacher aide, estimate the regression model explaining students' combined aptitude scores as a function of the presence of a teacher aide, TOTALSCORE \(=\gamma_{1}+\gamma_{2} A I D E+e\). Interpret the estimates. Based on this regression result, what do you conclude about the effect on learning of adding a teacher aide to the classroom?

d. Repeat part (c) using dependent variables READSCORE and MATHSCORE. Do you observe any differences?

Data From Part A:-

Using children who are in either a regular-sized class or a small class, estimate the regression model explaining students' combined aptitude scores as a function of class size, TOTALSCORE \(E_{i}=\) \(\beta_{1}+\beta_{2} S M A L L_{i}+e_{i}\). Interpret the estimates. Based on this regression result, what do you conclude about the effect of class size on learning?

Data From Part C:-

Using children who are in either a regular-sized class or a regular-sized class with a teacher aide, estimate the regression model explaining students' combined aptitude scores as a function of the presence of a teacher aide, TOTALSCORE \(=\gamma_{1}+\gamma_{2} A I D E+e\). Interpret the estimates. Based on this regression result, what do you conclude about the effect on learning of adding a teacher aide to the classroom?