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From the results of Problem 8.61, sx 13.966, sy 14.925 and r 0.7686. We now use (16), page 268, to obtain the equations of the regression lines. (a) or (b) or x x # rsx sy ( y y y # rsy sx (x x), y # y y), # x 69.0 y 70.9 69.0 (0.7686)(14.925) (x 13.966 70.9) 69.0), 70.9),

(0.7686) (13.966) (y 14.925 69.0)

where adj A ji is the transposed matrix of the cofactors of the aij in the determinant of A, and I is the n n unit matrix. If A is nonsingular, one may divide through by det A 6 0, and infer that A 1 1 adj A det A

There are two measures of the inheritance, the method inheritance factor (MIF) and the attribute inheritance factor (AIF). Md Ci Number of methods declared in a class i Mi Ci Number of methods inherited (and not overridden) in a class i Ma Ci Md Ci Mi Ci Number of methods that can be invoked in association with class i

8.63. Compute the standard errors of estimate (a) sy.x, (b) sy.x for the data of Problem 8.60. Use the results of Problem 8.61.

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This means that the unique solution of the linear system Y AX is 1 adj A Y X det A which is Cramer s rule in matrix form. The ordinary, determinant form is obtained by considering the rth row r 1; 2; . . . ; n of the matrix solution. Since the rth row of adj A is 1r 2r 3r . . . nr we have:

(a) (b) sy.x sx.y sy !1 sx !1 r2 r2 14.925 !1 13.966 !1 (0.7686)2 (0.7686)2 9.548 8.934

8.64. Fit a least-squares line to the data in Table 8-31 using (a) x as the independent variable, (b) x as the dependent variable. Graph the data and the least-squares lines using the same set of coordinate axes. Table 8-31 x y 3 2 5 3 6 4 8 6 9 5 11 8

1 y1 1r y2 2r y3 3r yn nr det A a11 a1 r 1 y1 a1 r 1 a1n 1 a21 a2 r 1 y2 a2 r 1 a2n det A a a y a a

8.66. Table 8-32 shows the final grades in algebra and physics obtained by 10 students selected at random from a large group of students. (a) Graph the data. (b) Find the least-squares line fitting the data, using x as the

The last equality may be veri ed by applying Laplace s theorem to the rth column of the given determinant.

independent variable. (c) Find the least-squares line fitting the data, using y as independent variable. (d) If a student receives a grade of 75 in algebra, what is her expected grade in physics (e) If a student receives a grade of 95 in physics, what is her expected grade in algebra Table 8-32 Algebra (x) Physics (y) 75 82 80 78 93 86 65 72 87 91 71 80 98 95 68 72 84 89 77 74

Ad Ci Number of attributes declared in a class i Ai Ci Number of attributes from base classes that are accessible in a class i Aa Ci Ad Ci Ai Ci Number of attributes that can be accessed in association with class i

8.67. Refer to Table 8-33. (a) Construct a scatter diagram. (b) Find the least-squares regression line of y on x. (c) Find the least-squares regression line of x on y. (d) Graph the two regression lines of (b) and (c) on the scatter diagram of (a). Table 8-33 Grade on First Quiz (x) Grade on Second Quiz (y) 6 8 5 7 8 7 8 10 7 5 6 8 10 10 4 6 9 8 7 6

For a linear system Y AX, with n n characteristic matrix A, it is of particular importance to investigate the excitations X that produce a proportionate response Y. Thus, letting Y X, where is a scalar, X AX or I A X O

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