Problem 11.12 The data in the next table were obtained in a forestry study of certain characteristics of a stand of pine trees including A, age of a particular pine stand; HD, the average height of dominant trees in feet; N, the number of pine trees per acre at age A; and MDBH, the average diameter measured at breast height (4.5 feet above ground). A theoretical analysis suggested that the following model is appropriate: Model (Y = MDBH|X1 = HD, X2 = A × N, X3 = HD/N).

(a) Fit this model to the data. Compute R2 and R2 a (the adjusted R-square–see Equation 11.52).

(b) Test the hypothesis H0 : β3 = 0 against H1 : β3

= 0. State your conclusions.

(c) Fit the reduced Model (Y = MDBH|X1 = HD, X2 = A × N). Compute R2 and R2 a for the reduced model.

(d) For each model prepare a table of fitted values, standard error of prediction, etc., similar in format to that of Table 11.4. Compare the full and reduced models by comparing the standard errors of prediction. Which model performs better? A HD N MDBH 19 51.5 500 7.0 14 41.3 900 5.0 11 36.7 650 6.2 13 32.2 480 5.2 13 39.0 520 6.2 12 29.8 610 5.2 18 51.2 700 6.2 14 46.8 760 6.4 20 61.8 930 6.4 17 55.8 690 6.4 13 37.3 800 5.4 21 54.2 650 6.4 11 32.5 530 5.4 19 56.3 680 6.7 17 52.8 620 6.7 15 47.0 900 5.9 16 53.0 620 6.9 16 50.3 730 6.9 14 50.5 680 6.9 22 57.7 480 7.9