1. (5) Test the regression relation between sales and the three predictor variables. State the hypotheses, test statistic and degrees of freedom, the p-value, the conclusion in words.

2. (5) Determine whether the linear regression model is appropriate by using the "usual" plots (scatterplot, residual plots, histogram/QQ plot). Explain in detail whether or not each assumption appears to be substantially violated.

3. (5)Prepare partial regression plot for each of the predictor variables. Do the plots suggest that the regression relationship in the fitted regression function are inappropriate for any of the predictor variables? Explain.

4. (5) Are there any outlying Y observations? (Show the diagnostic plot and test based on studentized residual).

5. (5) Are there any outlying X observations? (Show the diagnostic plot and test based on the Hat value).

6. (5) Are there any influential points? (Show the Cook's distance plot and test based on the Cook's distance)

7. Is there a serious multicollinearity problem?

a)(4) Include an appropriate scatterplot and correlation values between the explanatory variables.

b)(4) Judge by VIF, do you think there is a problem with multicollinearity? (Hint: VIP or tolerance)

c)(2) Compare your answers in parts a) and b). Are your conclusions the same or different? Please explain your answer.

8. Instead of removing variables, we are going to use the Ridge Regression to determine the parameter values.

a)(5) Make a ridge trace plot. What value of the parameter () do you believe is best? Explain your choice.

b) (5) Using the VIF factors, what value of the parameter do you believe should be used? (Hint: Look at both the graph and the printed numbers.) Explain your choice

c) (5) Report your model.

Cosmetics.csv:

y x1 x2 x3

12.85 5.6 5.6 3.8

11.55 4.1 4.8 4.8

12.78 3.7 3.5 3.6

11.19 4.8 4.5 5.2

9 3.4 3.7 2.9

9.34 6.1 5.8 3.4

13.8 7.7 7.2 3.8

8.79 4 4 3.8

8.54 2.8 2.3 2.9

6.23 3.2 3 2.8

11.77 4.2 4.5 5.1

8.04 2.7 2.1 4.3

5.8 1.8 2.5 2.3

11.57 5 4.6 3.6

7.03 2.9 3.2 4

0.27 0 0.2 2.7

5.1 1.4 2.2 3.8

9.91 4.2 4.3 4.3

6.56 2.4 2.2 3.7

14.17 4.7 4.7 3.4

8.32 4.5 4.4 2.7

7.32 3.6 2.9 2.8

3.45 0.6 0.8 3.4

13.73 5.6 4.7 5.3

8.06 3.2 3.3 3.6

9.94 3.7 3.5 4.3

11.54 5.5 4.9 3.2

10.8 3 3.6 4.6

12.33 5.8 5 4.5

2.96 3.5 3.1 3

7.38 2.3 2 2.2

8.68 2 1.8 2.5

11.51 4.9 5.3 3.8

1.6 0.1 0.3 2.7

10.93 3.6 3.8 3.8

11.61 4.9 4.4 2.5

17.99 8.4 8.2 3.9

9.58 2.1 2.3 3.9

7.05 1.9 1.8 3.8

8.85 2.4 2 2.4

7.53 3.6 3.5 2.4

10.47 3.6 3.7 4.4

11.03 3.9 3.6 2.9

12.31 5.5 5 5.5