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barcode font reporting services Curve Fitting, Regression, and Correlation in Software
CHAPTER 8 Curve Fitting, Regression, and Correlation Decoding QR Code 2d Barcode In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Generate QR Code ISO/IEC18004 In None Using Barcode creation for Software Control to generate, create QR Code image in Software applications. x0 for the population [i.e., yp # 2 x)2 >s2] # x (48) E(Y Z X x0)]. Decoding QR Code JIS X 0510 In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Making QR Code 2d Barcode In Visual C# Using Barcode encoder for .NET framework Control to generate, create Denso QR Bar Code image in .NET applications. denote the predicted mean value of y corresponding to x Then the statistic t ( y0 sy.x 21 yp)!n # [(x0 QRCode Encoder In VS .NET Using Barcode creation for ASP.NET Control to generate, create QRCode image in ASP.NET applications. QR Code Drawer In .NET Using Barcode creator for Visual Studio .NET Control to generate, create Denso QR Bar Code image in Visual Studio .NET applications. 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. QRCode Printer In Visual Basic .NET Using Barcode maker for .NET Control to generate, create QR Code 2d barcode image in .NET framework applications. Create DataMatrix In None Using Barcode creator for Software Control to generate, create Data Matrix image in Software applications. Sampling Theory of Correlation
GS1  12 Creator In None Using Barcode creator for Software Control to generate, create GS1  12 image in Software applications. Code 128A Creation In None Using Barcode printer for Software Control to generate, create Code 128 Code Set C image in Software applications. 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) Create USS Code 39 In None Using Barcode creator for Software Control to generate, create Code 39 Extended image in Software applications. UCC.EAN  128 Generator In None Using Barcode drawer for Software Control to generate, create USS128 image in Software applications. 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) Draw EAN 8 In None Using Barcode maker for Software Control to generate, create EAN / UCC  8 image in Software applications. Paint Barcode In Java Using Barcode drawer for BIRT Control to generate, create bar code image in Eclipse BIRT applications. 2. TEST OF HYPOTHESIS r 5 r0 u 0. Z 1 1 ln a 2 1
USS Code 39 Scanner In Visual Studio .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in VS .NET applications. Recognizing Code 3/9 In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. 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) Barcode Drawer In None Using Barcode generator for Office Excel Control to generate, create barcode image in Excel applications. Barcode Drawer In None Using Barcode drawer for Font Control to generate, create barcode image in Font applications. 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 Paint GTIN  12 In Java Using Barcode printer for Java Control to generate, create Universal Product Code version A image in Java applications. Barcode Printer In Java Using Barcode encoder for Eclipse BIRT Control to generate, create barcode image in BIRT reports applications. 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 315, 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 leastsquares 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. 85. The constant b ( y2 y1)>(x2 x1) is the slope of the line. Fig. 85 8.2. (a) Construct a straight line that approximates the data of Table 81. (b) Find an equation for this line. Table 81 x y 1 1 3 2 4 4 6 4 8 5 9 7 11 8 14 9 Fig. 86

