Younger (1979, p. 533) presents data from a sample of 12 discount department stores that advertise on television, radio, and in the newspapers. The variables x1, x2, and x3 represent the respective amounts of money spent on these advertising activities during a certain month while y gives the store’s revenues during that month. The data are given in Table 9.2. Complete the following tasks using multiple regression.

(a) Give the theoretical model along with the relevant assumptions.

(b) Give the fitted model, i.e., repeat

(a) substituting the estimates for the unknown parameters.

(c) Test H0 : β2 = 0 versus HA : β2 = 0 at α = 0.05.

(d) Test the hypothesis H0 : β1 =β2 =β3 = 0.

(e) Give a 99% confidence interval for β2.

Table 9.2: Younger’s advertising data.
Obs. y x1 x2 x3 Obs. y x1 x2 x3 1 84 13 5 2 7 34 12 7 2 2 84 13 7 1 8 30 10 3 2 3 80 8 6 3 9 54 8 5 2 4 50 9 5 3 10 40 10 5 3 5 20 9 3 1 11 57 5 6 2 6 68 13 5 1 12 46 5 7 2

(f) Test whether the reduced model yi = β0 +β1xi1 +εi is an adequate explanation of the data as compared to the full model.
(g) Test whether the reduced model yi = β0 +β1xi1 +εi is an adequate explanation of the data as compared to the model yi =β0+β1xi1 +β2xi2+εi.
(h) Write down the ANOVA table for the ‘full’ model used in (g).
(i) Construct an added variable plot for adding variable x3 to a model that already contains variables x1 and x2. Interpret the plot.
(j) Compute the sample partial correlation ry3·12. What does this value tell you?