Theme: Building 95% confidence bands around the simple linear regression models. Download the data 'GRADES' from the package 'PASWR2.' Let us practice simple linear regression and 95% bands. Throw in some correlation.

What is the dataset about? The setting is a large public university. As you all know, admission officers look at your SAT and/or ACT scores. The belief is that the higher you score is in any of the entrance exams, the better your performance is in college in terms of higher GPA. Is this true? The college admissions board takes a random sample 200 freshmen in their second semester. For each freshmen in the sample, let X = sat and Y = gpa in their first semester. The board is inclining to abandon sat requirement for college admission. The data are given in GRADES. We want you to build a causal model connecting Y with X linearly.

1. What is the dimension of the data? ( 1 point )

2. Show the top 20 rows of the data. (1 point )

3. Get the summary statistics of the data. (2 points )

4. Write the population Simple Linear Regression Model. State the assumptions of the model. ( 2+2 points )

5. Fit the model and show the output. ( 2+2 points )

6. Write the prediction equation. (2 points )

7. Interpret the model. ( 2 points )

8. Check the assumptions of the model. ( 4 points )

9. Check the significance of the regression coefficient associated with sat. ( 2 points )

10. Do you think that sat has a significant impact on gpa and why? (2 points )

11. Calculate the correlation coefficient between sat and gpa. Test its significance. ( 2+2 points )

12. Report R2 ( R square ) and assess its significance. ( 2+2 points )

13. Obtain a 95% confidence band for the regression line in Question 5. Show the top six rows of the output. ( 2+2 points )

14. Obtain a 95% prediction band for the regression line in Question 5. Show the top six lines of the output. ( 2+2 points )

15. Put everything together in a graph (scatter plot, best line, 95% Confidence band, 95%Prediction Band, legend). Comments on your graph. ( 2+2+6+6+2+2 points )