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barcode font reporting services g(x) having property (37) is called a leastsquares regression curve of Y on X. We have in Software
g(x) having property (37) is called a leastsquares regression curve of Y on X. We have QR Code Scanner In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. QRCode Printer In None Using Barcode creator for Software Control to generate, create Denso QR Bar Code image in Software applications. Theorem 81 If X and Y are random variables having joint density function or probability function f(x, y), then there exists a leastsquares regression curve of Y on X, having property (37), given by y g(x) E(Y Z X x) (38) Reading QR Code 2d Barcode In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. QR Maker In Visual C# Using Barcode drawer for .NET framework Control to generate, create Denso QR Bar Code image in .NET applications. provided that X and Y each have a variance that is finite. Note that E(Y Z X x) is the conditional expectation of Y given X x, as defined on page 82. Similar remarks can be made for a leastsquares regression curve of X on Y. In that case, (37) is replaced by E5[X h(Y)]26 a minimum QR Creator In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications. Creating QRCode In Visual Studio .NET Using Barcode creator for VS .NET Control to generate, create QR Code JIS X 0510 image in .NET applications. and (38) is replaced by x h( y) E(X Z Y y). The two regression curves y g(x) and x h( y) are different in general. An interesting case arises when the joint distribution is the bivariate normal distribution given by (49), page 117. We then have the following theorem: Theorem 82 If X and Y are random variables having the bivariate normal distribution, then the leastsquares regression curve of Y on X is a regression line given by y mY sY r ra x mX sX b (39) Painting Quick Response Code In Visual Basic .NET Using Barcode encoder for Visual Studio .NET Control to generate, create QRCode image in .NET framework applications. Drawing GS1  13 In None Using Barcode encoder for Software Control to generate, create GTIN  13 image in Software applications. where
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Create GTIN  128 In None Using Barcode generation for Software Control to generate, create EAN128 image in Software applications. Encoding UPC Code In None Using Barcode encoder for Software Control to generate, create UPC Code image in Software applications. (40) Create UPC Shipping Container Symbol ITF14 In None Using Barcode printer for Software Control to generate, create UCC  14 image in Software applications. Recognize GS1  12 In .NET Framework Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications. represents the population correlation coefficient. We can also write (39) as y where mY b b(x sXY s2 X mX) (41) (42) Bar Code Maker In .NET Using Barcode printer for Reporting Service Control to generate, create barcode image in Reporting Service applications. Drawing GS1 128 In Java Using Barcode encoder for Java Control to generate, create GS1128 image in Java applications. Similar remarks can be made for the leastsquares regression curve of X on Y, which also turns out to be a line [given by (39) with X and Y, x and y, interchanged]. These results should be compared with corresponding ones on page 268. Read Code 128 Code Set A In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Code 39 Reader In VB.NET Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications. CHAPTER 8 Curve Fitting, Regression, and Correlation
Create UCC  12 In None Using Barcode generator for Excel Control to generate, create EAN128 image in Excel applications. Printing UCC.EAN  128 In ObjectiveC Using Barcode maker for iPad Control to generate, create EAN / UCC  13 image in iPad applications. In case f (x, y) is not known, we can still use the criterion (37) to obtain approximating regression curves for the population. For example, if we assume g(x) a bx, we obtain the leastsquares regression line (39), where a, b are given in terms of the (unknown) parameters mX, mY, sX, sY, r. Similarly if g(x) a bx gx2, we can obtain a leastsquares regression parabola, etc. See Problem 8.39. In general, all of the remarks made on pages 266 to 271 for samples are easily extended to the population. For example, the standard error of estimate in the case of the population is given in terms of the variance and correlation coefficient by s2 Y.X which should be compared with (25), page 269. s2 (1 Y r2) (43) Probability Interpretation of Correlation
From the above remarks it is clear that a population correlation coefficient should provide a measure of how well a given population regression curve fits the population data. All the remarks previously made for correlation in a sample apply as well to the population. For example, if g(x) is determined by (37), then E[(Y # Y )2] E[(Y Yest)2] E[(Yest # Y )2] (44) # where Yest g(X ) and Y E(Y ). The three quantities in (44) are called the total, unexplained, and explained variations, respectively. This leads to the definition of the population correlation coefficient r, where r2 explained variation total variation E[(Yest E[(Y # Y )2] # Y )2] (45) For the linear case this reduces to (40). Results similar to (31) through (34) can also be written for the case of a population and linear regression. The result (45) is also used to define r, in the nonlinear case. Sampling Theory of Regression
The regression equation y a bx is obtained on the basis of sample data. We are often interested in the corresponding regression equation y a bx for the population from which the sample was drawn. The following are some tests concerning a normal population. To keep the notation simple, we shall follow the common convention of indicating values of sampling random variables rather than the random variables themselves. 1. TEST OF HYPOTHESIS B 5 b. To test the hypothesis that the regression coefficient b is equal to some specified value b, we use the fact that the statistic t b b !n sy.x >sx 2 (46) has Student s distribution with n 2 degrees of freedom. This can also be used to find confidence intervals for population regression coefficients from sample values. See Problems 8.43 and 8.44. 2. TEST OF HYPOTHESES FOR PREDICTED VALUES. Let y0 denote the predicted value of y corresponding to x x0 as estimated from the sample regression equation, i.e., y0 a bx0. Let yp denote the predicted value of y corresponding to x x0 for the population. Then the statistic t ( y0 sy.x 2n 1 yp)!n [n(x0 2 x)2 >s2] # x (47) has Student s distribution with n 2 degrees of freedom. From this, confidence limits for predicted population values can be found. See Problem 8.45. 3. TEST OF HYPOTHESES FOR PREDICTED MEAN VALUES. Let y0 denote the predicted value of y corresponding to x x0 as estimated from the sample regression equation, i.e., y0 a bx0. Let yp #

