Question 6 A random sample of 1000 students from a large university was interviewed about their high school SAT scores (X) and their current university GPAs (Y). The students were classified into two groups: those with SAT scores greater than or equal to the national average (X = 1) and those with SAT scores below the national average (X = 0). The average GPA was calculated for each group. No of observations Average GPA SAT below average SAT above average 400 600 2.9 3.2 a) What is the estimate of the population mean GPA of students with SAT scores below the national average? What is the estimator used to estimate this mean? For both the estimate and the estimator, provide either a numerical answer or a formula. Which of the two - the estimator and estimate - is a random variable, and which is a realized value of a random variable? b) What is the expected GPA for all students in the university? (Hint: use the "Law of Iter- ated Expectations. The two given average GPAs are estimates of two conditional means, one conditional on below average SAT and the other conditional on above average SAT.) c) Assume that the variance of a single student's average GPA equals o?, independently of his or her SAT score. Is the mean GPA estimator more or less precise for students with below average SAT than for those with above average SAT scores? By how much? (Note: the precision of a random variable, V, is the inverse of its variance.) d) If a student scored below the national average on her SAT, does that mean that her expected GPA is below the expected GPA of a student who scored above average? Does it mean that her actual - not expected - GPA is necessarily below that of a student who scored above average? Explain. Question 6 A random sample of 1000 students from a large university was interviewed about their high school SAT scores (X) and their current university GPAs (Y). The students were classified into two groups: those with SAT scores greater than or equal to the national average (X = 1) and those with SAT scores below the national average (X = 0). The average GPA was calculated for each group. No of observations Average GPA SAT below average SAT above average 400 600 2.9 3.2 a) What is the estimate of the population mean GPA of students with SAT scores below the national average? What is the estimator used to estimate this mean? For both the estimate and the estimator, provide either a numerical answer or a formula. Which of the two - the estimator and estimate - is a random variable, and which is a realized value of a random variable? b) What is the expected GPA for all students in the university? (Hint: use the "Law of Iter- ated Expectations. The two given average GPAs are estimates of two conditional means, one conditional on below average SAT and the other conditional on above average SAT.) c) Assume that the variance of a single student's average GPA equals o?, independently of his or her SAT score. Is the mean GPA estimator more or less precise for students with below average SAT than for those with above average SAT scores? By how much? (Note: the precision of a random variable, V, is the inverse of its variance.) d) If a student scored below the national average on her SAT, does that mean that her expected GPA is below the expected GPA of a student who scored above average? Does it mean that her actual - not expected - GPA is necessarily below that of a student who scored above average? Explain