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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8.29. Prove the result (17), page 268.
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The regression line of y on x is y Similarly, the regression line of x on y is x Then c bd dy where d rsy rsx a s ba s b
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bx where b
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rsy sx
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rsx sy r2
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8.30. Use the result of Problem 8.29 to find the linear correlation coefficient for the data of Problem 8.11.
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From Problem 8.11(b) and 8.11(c), respectively, b Then r2 484 1016 bd a 0.476 d 484 467 or r 1.036 0.7027
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484 484 ba b 1016 467
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agreeing with Problems 8.26(b) and 8.28.
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8.31. Show that the linear correlation coefficient is given by n a xy r B
In Problem 8.27 it was shown that (1) r B But a (x x)(y # y) # a xryr Q a xr2 R Q a yr2 R a (xy a xy a xy since x # (gx)>n and y # a (x (gy)>n. x)2 # a (x a x2 and Then (1) becomes a xy r B S a x2 Q a xR Q a yR >n
Q a xR Q a yR
Sn a x2
Q a xR T Sn a y2
Q a yR T
a (x B xy # ny x ## S a (x x y) ## nx y ##
x)(y # x)2 T S a (y # a xy a xy
y) # y)2 T # y ax # nx y ##
xy # nx y ##
x ay # nx y ##
Q a xR Q a yR n
Similarly,
2xx # 2Q a xR n y)2 #
x2) #
2x a x # ax
nx2 # Q a xR n
Q a xR n
a (y
a y2
Q a yR n
n a xy
Q a xR Q a yR
Q a xR >nT S a y2
Q a yR >nT
Sn a x2
Q a xR T Sn a y2
Q a yR T
CHAPTER 8 Curve Fitting, Regression, and Correlation
8.32. Use the formula of Problem 8.31 to obtain the linear correlation coefficient for the data of Problem 8.11.
From Table 8-4, n a xy r B Sn a x2 Q a xR Q a yR
Q a xR T Sn a y2 (12)(54,107)
Q a yR T 0.7027
(800)(811) (811)2]
![(12)(53,418) as in Problems 8.26(b), 8.28, and 8.30.
(800)2][(12)(54,849)
Generalized correlation coefficient 8.33. (a) Find the linear correlation coefficient between the variables x and y of Problem 8.16. (b) Find a nonlinear correlation coefficient between these variables, assuming the parabolic relationship obtained in Problem 8.16. (c) Explain the difference between the correlation coefficients obtained in (a) and (b). (d) What percentage of the total variation remains unexplained by the assumption of parabolic relationship between x and y
(a) Using the calculations in Table 8-9 and the added fact that gy2 n a xy r B Sn a x2 Q a xR Q a yR
290.52, we find
Q a xR T Sn a y2 (8)(230.42)
Q a yR T 0.3743
(42.2)(46.4) (46.4)2]
![(8)(291.20) (b) From Table 8-9, y # From Table 8-10, (gy)>n
(42.2)2][(8)(290.52) 5.80. Then a (y a (yest y)2 #
(46.4)>8
Total variation
21.40 y)2 #
Explained variation Therefore, r2 explained variation total variation
21.02 r 0.9911
21.02 21.40
0.9822 and
(c) The fact that part (a) shows a linear correlation coefficient of only 0.3743 indicates practically no linear relationship between x and y. However, there is a very good nonlinear relationship supplied by the parabola of Problem 8.16, as is indicated by the fact that the correlation coefficient in (b) is very nearly 1. (d) Unexplained variation Total variation 1 r2 1 0.9822 0.0178
Therefore, 1.78% of the total variation remains unexplained. This could be due to random fluctuations or to an additional variable that has not been considered.
8.34. Find (a) sy and (b) sy.x for the data of Problem 8.16.
(a) From Problem 8.33(b), g( y sy y)2 # a (y n 21.40. Then the standard deviation of y is y)2 # 21.40 A 8 1.636 or 1.64
(b) First method Using (a) and Problem 8.33(b), the standard error of estimate of y on x is sy.x sy !1 r2 1.636 !1 (0.9911)2 0.218 or 0.22
CHAPTER 8 Curve Fitting, Regression, and Correlation
Second method Using Problem 8.33, sy.x a( y B n yest)2 unexplained variation n B 21.40 A 8 21.02 0.218 or 0.22
Third method Using Problem 8.16 and the additional calculation gy2 sy.x a y2 B a ay b a xy n
290.52, we have c a x2y 0.218 or 0.22
8.35. Explain how you would determine a multiple correlation coefficient for the variables in Problem 8.19.
Since z is determined from x and y, we are interested in the multiple correlation coefficient of z on x and y. To obtain this, we see from Problem 8.19 that Unexplained variation
2 a (z zest) (64 64.414)2
(68 nz2 #
65.920)2
Total variation
2 #2 a (z z) az 48,139 12(62.75)2
888.25 629.37
Explained variation Then
Multiple correlation coefficient of z on x and y B explained variation total variation 629.37 A 888.25 0.8418
It should be mentioned that if we were to consider the regression of x on y and z, the multiple correlation coefficient of x on y and z would in general be different from the above value.
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