DS 12.11.1 gives the times taken by a class of n = 14 students to finish a math test together with the math SAT scores of the students. Perform the following linear regression analysis with time as the dependent variable and SAT as the explanatory variable.
(a) Calculate the point estimates β^0, β^1, and σ^2.
(b) Calculate the sample correlation coefficient r.
(c) Calculate a two-sided 959? confidence interval for the slope parameter β1.
(d) Is there any evidence that the time taken to finish the test depends upon the SAT score?
(e) How does the predicted time taken to finish the test change as the SAT score increases by 10 points?
(f) What is the predicted time for a student with an SAT score of 550? Construct a two-sided 95% confidence interval for the expected time for a student with an SAT score of 550. Construct a two-sided 95% prediction interval for the time taken by a student with an SAT score of 550.