barcode font reporting services Curve Fitting, Regression, and Correlation in Software

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CHAPTER 8 Curve Fitting, Regression, and Correlation
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(b) From the figure it is seen that of the 12 data points, 7 fall between the lines while 3 appear to lie on the lines. Further examination using the last line in Table 8-14 reveals that 2 of these 3 points lie between the lines. Then the required percentage is 9>12 75%. Another method From the last line in Table 8-14, y yest lies between 1.28 and 1.28 (i.e., sy.x) for 9 points (x, y). Then the required percentage is 9>12 75%. If the points are normally distributed about the regression line, theory predicts that about 68% of the points lie between the lines. This would have been more nearly the case if the sample size were large.
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A better estimate of the standard error of estimate of the population from which the sample heights !n>(n 2)sy.x !12>10(1.28) 1.40 inches. were taken is given by ^y.x s
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The linear correlation coefficient 8.24. Prove that g( y y)2 yest)2 g( y #
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Squaring both side of y a( y y # y)2 # (y a( y yest) yest)2
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g( yest
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( yest
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y) and then summing, we have # y)2 # 2 a( y yest)( yest y) #
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a ( yest
The required result follows at once if we can show that the last sum is zero. In the case of linear regression this is so, since a (y yest)(yest y) # a (y a a (y 0 because of the normal equations g(y a bx) 0, gx(y a bx) 0. The result can similarly be shown valid for nonlinear regression using a least-squares curve given by yest a0 a1x a2x2 c anxn. a a bx)(a bx) bx y) # a bx) y a (y # a bx)
b a x(y
8.25. Compute (a) the explained variation, (b) the unexplained variation, (c) the total variation for the data of Problem 8.11.
We have y 67.58 from Problem 8.12 (or from Table 8-4, since y # # from Table 8-14 we can construct Table 8-15. Table 8-15 yest y # 0.82 1.77 0.13 1.30 0.61 2.25 c 1.56 0.34 0.61 0.13 1.08 2.04 811>12 67.58). Using the values yest
(a) Explained variation (b) Unexplained variation
g(yest g(y
y)2 # yest)2
( 0.82)2 ns2 y.x
(2.04)2
19.70, from Problem 8.22.
(c) Total variation g(y y)2 19.22 19.70 38.92, from Problem 8.24. # The results in (b) and (c) can also be obtained by direct calculation of the sum of squares.
8.26. Find (a) the coefficient of determination, (b) the coefficient of correlation for the data of Problem 8.11. Use the results of Problem 8.25.
(a) Coefficient of determination (b) Coefficient of correlation r r r2 explained variation total variation !0.4938 0.7027. 19.22 38.92 0.4938.
Since the variable yest increases as x increases, the correlation is positive, and we therefore write 0.7027, or 0.70 to two significant figures.
CHAPTER 8 Curve Fitting, Regression, and Correlation
8.27. Starting from the general result (30), page 270, for the correlation coefficient, derive the result (34), page 271 (the product-moment formula), in the case of linear regression.
The least-squares regression line of y on x can be written yest a b x x, and yrest yest y. Then, using yr gxryr> gxr2, xr # # explained variation total variation a b xr a yr2
bx or yrest bxr, where y y, we have # a yrest a yr2 Q a xryrR
a ( yest a( y
y)2 # y)2 #
b2 a xr2 a yr2
and so
2 a xryr a xr a xr2 a yr2
a xr2 a yr2
a xryr
2 2 $a xr a yr
However, since gxryr is positive when yest increases as x increases, but negative when yest decreases as x increases, the expression for r automatically has the correct sign associated with it. Therefore, the required result follows.
8.28. By using the product-moment formula, obtain the linear correlation coefficient for the data of Problem 8.11.
The work involved in the computation can be organized as in Table 8-16. Then a xryr B agreeing with Problem 8.26(b). Qa xr2 R Q a yr2 R 40.34 !(84.68)(38.92)
Table 8-16 x 65 63 67 64 68 62 70 66 68 67 69 71 gx x # 800 800>12 66.7 gy y # y 68 66 68 65 69 66 68 65 71 67 68 70 811 811>12 67.6 xr x x # 1.7 3.7 0.3 2.7 1.3 4.7 3.3 0.07 1.3 0.3 2.3 4.3 yr y y # 0.4 1.6 0.4 2.6 1.4 1.6 0.4 2.6 3.4 0.6 0.4 2.4 xr2 2.89 13.69 0.09 7.29 1.69 22.09 10.89 0.49 1.69 0.09 5.29 18.49 gxr2 84.68 xryr 0.68 5.92 0.12 7.02 1.82 7.52 1.32 1.82 4.42 0.18 0.92 10.32 gxryr 40.34 yr2 0.16 2.56 0.16 6.76 1.96 2.56 0.16 6.76 11.56 0.36 0.16 5.76 gyr2 38.92
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