Question - SAT versus ACT, continued. Refer to the previous exercise. Find the predicted value of ACT for each observation in the data set.
(a) What is the mean of these predicted values? Compare it with the actual mean of the ACT scores.
(b) Compare the standard deviation of the predicted values with the standard deviation of the actual ACT scores. If least-squares regression is used to predict ACT scores for a large number of students such as these, the average predicted value will be accurate but the variability of the predicted scores will be too small.
(c) Find the SAT score for a student who is one standard deviation above the mean (z = (x - x)/s = 1). Find the predicted ACT score and standardize this score. (Use the means and standard deviations from this set of data for these calculations.)
(d) Repeat part (c) for a student whose SAT score is one standard deviation below the mean (z = -1).
(e) What do you conclude from parts (c) and (d)? Perform additional calculations for different z's if needed.