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6 7 The equations of the two lines are y 11 11 x and x x 7, y 5. Therefore, the lines intersect in point (7, 5). 1 2 3 2 y.
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12 into the regression line of y on x, y 3 into the regression line of x on y, x
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8.7. Prove that a least-squares line always passes through the point (x, y). # #
Case 1 x is the independent variable. The equation of the least-squares line is A normal equation for the least-squares line is Dividing both sides of (2) by n gives Subtracting (3) from (1), the least-squares line can be written (4) y y # b(x x) # (1) (2) (3) y a bx
a y an b a x y a bx # #
which shows that the line passes through the point (x, y). # #
CHAPTER 8 Curve Fitting, Regression, and Correlation
Case 2
y is the independent variable. Proceeding as in Case 1 with x and y interchanged and the constants a, b, replaced by c, d, respectively, we find that the least-squares line can be written (5) x x # d(y y) #
which indicates that the line passes through the point (x, y). # # Note that, in general, lines (4) and (5) are not coincident, but they intersect in (x, y). # #
8.8. Prove that the least-squares regression line of y on x can be written in the form (8), page 267.
We have from (4) of Problem 8.7, y y # b(x x). From the second equation in (5), page 267, we have # n a xy (1) Now a (x b n a x2 x)2 # a (x ax ax ax
2 2 2 2
Q a xR Q a yR Q a xR x2) # # ax nx2 #
2xx # 2x a x # 2nx2 # nx2 #
2 1 n Q a xR
1 2 n Sn a x Also a (x x)( y # y) # a (xy a xy a xy a xy a xy 1 n Sn a xy Therefore, (1) becomes b a (x a (x x)(y #
Q a xR T xy # x ay # nx y ## nx y ## Q a xR Q a yR n Q a xR Q a yR T yx # x y) ## y ax # ny x ## ## ax y nx y ##
y) #
from which the result (8) is obtained. Proof of (12), page 268, follows on interchanging x and y.
8.9. Let x
h, y
k, where h and k are any constants. Prove that n a xy Q a xR Q a yR Q a xR
n a xryr n a xr2
Q a xrR Q a yrR Q a xrR
b n a x2
From Problem 8.8 we have n a xy b
Q a xR Q a yR Q a xR
a (x a (x
x)(y # x)2 #
y) #
n a x2
CHAPTER 8 Curve Fitting, Regression, and Correlation
xr h, y yr k, we have x # xr # y) # h, y # a (xr xr # k yr)
Now if x
Thus
a (x a (x
x)(y # x)2 #
xr)(yr xr)2
a (xr n a xryr n a xr2
Q a xrR Q a yrR Q a xrR
The result is useful in developing a shortcut for obtaining least-squares lines by subtracting suitable constants from the given values of x and y (see Problem 8.12).
8.10. If, in particular, h
x, k #
y in Problem 8.9, show that # b a xryr a xr2
This follows at once from Problem 8.9 since a xr and similarly gyr 0. a (x x) # ax nx # 0
8.11. Table 8-3 shows the respective heights x and y of a sample of 12 fathers and their oldest sons. (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.
Table 8-3 Height x of Father (inches) Height y of Son (inches) 65 68 63 66 67 68 64 65 68 69 62 66 70 68 66 65 68 71 67 67 69 68 71 70
(a) The scatter diagram is obtained by plotting the points (x, y) on a rectangular coordinate system as shown in Fig. 8-9.
Fig. 8-9
CHAPTER 8 Curve Fitting, Regression, and Correlation
(b) The regression line of y on x is given by y equations ay a xy a an a ax
bx, where a and b are obtained by solving the normal b ax b a x2
The sums are shown in Table 8-4, and so the normal equations become 12a 800a from which we find a shown in Fig. 8-9. Another method Q a yR Q a x2 R a n a x2 Q a xR Q a xR Q a xyR
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