We introduced Tennessee's Project STAR (Student/Teacher Achievement Ratio) in Exercise 2.22. The data file is star5_small. [The data file star 5 contains more observations and more variables.] Three types of classes were considered: small classes [SMALL \(=1]\), 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 regular classes with no teacher aide. Compute the sample mean and standard deviation for student math scores, MATHSCORE, in regular classes with a teacher aide. Which type of class had the higher average score? What is the difference in sample average scores for regular-sized classes versus regular sized classes with a teacher aide? Which type of class had the higher score standard deviation?

b. Consider students only in regular sized classes without a teacher aide and regular sized classes with a teacher aide. Estimate the regression model MATHSCORE \(=\beta_{1}+\beta_{2} A I D E+e\). 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 without a teacher aide and a regular-sized class with a teacher aide. 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 regular-sized classes with a teacher aide, 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 three points, or more, higher for students in regular-sized classes with a teacher aide versus the alternative that the difference is less than three points, 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?