22. (Hald's regression data). We present below a small set of data on heat evolved during the hardening of Portland cement and four variables that may be related to it (Woods et al. (1932), pp. 635-649). The sample size ( n) is 13. The regressor variables (in percent of the weight) are x 1 = calcium aluminate (3Cao.Ab03), x2 = tricalcium silicate (3CaO.Si02), x 3 = tetracalcium alumino ferrite (4Ca0.Ab03.Fe203), and X4 = dicalcium X1 X2 X3 X4 Y 7 26 6 60 78.6 1 29 15 52 74.3 11 56 8 20 104.3 11 31 8 47 87.6 7 52 6 33 95.9 11 55 9 22 109.2 3 7117 6 102.7 1 31 22 44 72.5 2 54 18 22 93.1 21 47 4 26 115.9 1 40 23 34 83.8 11 66 9 12 113.3 10 68 8 12 109.4 silicate (2CaO.Si02 ). The response variable is y = total calories given off during hardening per gram of cement after 180 days. Usually such a data set is analyzed using normal linear regression model of the form where pis the number of regressor variables in the model, {30 , {31 , ... {3p are unknown parameters, and Ei 's are independent errors having a N(O, a 2 ) distribution. There are a number of possible models depending on which regressor variables are kept in the model. Analyze the data and choose one from this set of possible models using

(a) BIC,

(b) AIBF of the full model relative to all possible models.