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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x0 for the population [i.e., yp # 2 x)2 >s2] # x (48) E(Y Z X x0)].
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denote the predicted mean value of y corresponding to x Then the statistic t ( y0 sy.x 21 yp)!n # [(x0
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has Student s distribution with n 2 degrees of freedom. From this, confidence limits for predicted mean population values can be found. See Problem 8.46.
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Sampling Theory of Correlation
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We often have to estimate the population correlation coefficient r from the sampling correlation coefficient r or to test hypotheses concerning r. For this purpose we must know the sampling distribution of r. In case r 0, this distribution is symmetric and a statistic having Student s distribution can be used. For r 2 0, the distribution is skewed. In that case a transformation due to Fisher produces a statistic which is approximately normally distributed. The following tests summarize the procedures involved. 1. TEST OF HYPOTHESIS r 5 0. Here we use the fact that the statistic t has Student s distribution with n r!n !1 2 r2 (49)
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2 degrees of freedom. See Problems 8.47 and 8.48. Here we use the fact that the statistic r b r 1.1513 log10 a 1 1 r b r (50)
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2. TEST OF HYPOTHESIS r 5 r0 u 0. Z 1 1 ln a 2 1
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is approximately normally distributed with mean and standard deviation given by mz 1 1 ln a 2 1 r0 b r0 1.1513 log10 a 1 1 r0 b, r0 sZ 1 !n (51)
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These facts can also be used to find confidence limits for correlation coefficients. See Problems 8.49 and 8.50. The transformation (50) is called Fisher s Z transformation. 3. SIGNIFICANCE OF A DIFFERENCE BETWEEN CORRELATION COEFFICIENTS. To determine whether two correlation coefficients r1 and r2, drawn from samples of sizes n1 and n2, respectively, differ significantly from each other, we compute Z1 and Z2 corresponding to r1 and r2 using (50). We then use the fact that the test statistic Z1 Z2 mZ Z 1 2 (52) z sZ Z
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where
mZ ,
2s2 Z
s2 Z
1 A n1 3 n2
(53)
is normally distributed. See Problem 8.51.
Correlation and Dependence
Whenever two random variables X and Y have a nonzero correlation coefficient r, we know (Theorem 3-15, page 81) that they are dependent in the probability sense (i.e., their joint distribution does not factor into their marginal distributions). Furthermore, when r 2 0, we can use an equation of the form (39) to predict the value of Y from the value of X. It is important to realize that correlation and dependence in the above sense do not necessarily imply a direct causal interdependence of X and Y. This is shown in the following examples.
CHAPTER 8 Curve Fitting, Regression, and Correlation
EXAMPLE 8.1 Let X and Y be random variables representing heights and weights of individuals. Here there is a direct interdependence between X and Y. EXAMPLE 8.2 If X represents teachers salaries over the years while Y represents the amount of crime, the correlation coefficient may be different from zero and we may be able to find a regression equation predicting one variable from the other. But we would hardly be willing to say that there is a direct interdependence between X and Y.
SOLVED PROBLEMS
The least-squares line 8.1. A straight line passes through the points (x1, y1) and (x2, y2). Show that the equation of the line is
The equation of a line is y a
y2 ax
y1 x1 b(x
bx. Then since (x1, y1) and (x2, y2) are points on the line, we have y1 a bx1, y2 a bx2
Therefore, (1) (2) y y2 y1 y1 (a (a bx) bx2) (a (a bx1) bx1) b(x b(x2 x1) x1)
Obtaining b (y2 y1)>(x2 x1) from (2) and substituting in (1), the required result follows. The graph of the line PQ is shown in Fig. 8-5. The constant b ( y2 y1)>(x2 x1) is the slope of the line.
Fig. 8-5
8.2. (a) Construct a straight line that approximates the data of Table 8-1. (b) Find an equation for this line. Table 8-1 x y 1 1 3 2 4 4 6 4 8 5 9 7 11 8 14 9
Fig. 8-6
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