11.11 Properties of the Least Squares Estimators for the Multiple Linear Regression Model 527 11.57 Florida morbidity statistics for the decade ending in 1976 show that infectious hepatitis had the incidence rates shown in the accompanying table (in cases per 100,000 population). x y 1967 10.5 1968 18.5 1969 22.6 1970 27.2 1971 31.2 1972 33.0 1973 44.9 1974 49.4 1975 35.0 Question 1976 27.6 1. -a Letting Y denote incidence rate and x denote the coded year (-9 for 1967, -7 for 1968, through 9 for 1976), fit the model Y = Bo + Bix + &. 2. b For the same data, fit the model Y = Bo + Bix + B2x2 + . 11.58 a Calculate SSE and $2 for Exercise 11.2. Use the matrix approach. b Fit the model suggested in Exercise 11.2 for the relationship between audited values and book values by using matrices. We can simplify the computations by defining and fitting the model Y = Bo + Bix* + E. Fit this latter model and calculate SSE. Compare your answer with the SSE calculation in (a).LE 11.11 Solve Example 11.1 by using matrix operations. Solution From the data given in Example 11.1, we see that Ox *1 -2- - 1 Y = and X = - O N It follows that x' X = [8 10] x x = Thus, B = ( X'X ) 'X' Y = 1/5 , 90][9]= or Bo = 1 and B, = .7. Thus, y =1+.7x just as in Example 11.1. . 11.12 Fit a parabola to the data of Example 11.1, using the model Y = Bot Bix + B2x2 + E. Solution The X matrix for this example differs from that of Example 11.1 1 only by the addition of a third column corresponding to x . (Notice that x] = x, X2 = x2, and k = 2 in the notation of the general linear model.) Thus, X 7 2 Y = X = (The three variables, Xo, x, and x , are shown above their respective columns in the X matrix.) Thus, for the first measurement, y = 0, xo = 1, x = -2, and x = 4; and for the second measurement, y = 0, * = 1, x = -1, and x2 = 1. Succeeding rows of the Y and X matrices are obtained in a similar manner