The article “A Statistical Analysis of the Notch Toughness of 9% Nickel Steels Obtained from Production Heats” (J. of Testing and Eval., 1987: 355–363) reports on the results of a multiple regression analysis relating Charpy v-notch toughness y (joules) to the following variables: x₁ =  plate thickness (mm), x₂ = carbon content (%), x₃ = manganese content (%), x₄ = phosphorus content (%)x₅ = sulphur content (%), x₆   = silicon content (%),  x₇ = nickel content (%), yield strength (Pa), and  x₉ =. tensile strength (Pa).

a. The best possible subsets involved adding variables in the order x₅, x₈, x₆, x₃, x₂, x₇, x₉, x₁, and x₄. The values of , MSE_k, and C_k R_k²  MSEk, and Ck  are as follows:

  

                

         
 

Which model would you recommend? Explain the rationale for your choice.

  

b. The authors also considered second-order models involving predictors and . Information on the best such models starting with the variables x₂, x₃, x₅, x₆, x₇, and x₈ is as follows (in going from the best four-predictor model to the best five-predictor model, x₈ was deleted and both x₂x₆ and x₇x₈ were entered, and x8 was reentered at a later stage):

  

                

  

Which of these models would you recommend, and why? 

X = plate thickness (mm), x, = carbon content (%), x, = manganese content (%), x = phosphorus content (%) X5 = sulphur content (%), x, = silicon content (%), x, = nickel content (%), xg = tensile strength (Pa) a. The best possible subsets involved adding variables in yield strength (Pa), and x, = . the order x3, xg, X6, X3, Xz, X7, Xg, X1, and x4. The values of R, MSE,, and C, are as follows: No. of Predictors 1 2 3 4 .354 .453 .511 .550 MSE, C 2295 1948 1742 1607 314 173 89.6 35.7