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A textbook of Computer Based Numerical and Statiscal Techniques part 44

A textbook of Computer Based Numerical and Statiscal Techniques part 44. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 416 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES And the equation of the line of regression of y on x is y - y r. x - x nx . ii Let m1 and m2 be the slopes of i and ii respectively. Then m1 and m2 -Hx r.ox Therefore m1 - m2 tan 0 1--- 1 m1m2 rHy L_ rHx Hx i fe x 2 1 - r 2 HxHy Proved. Example 9. The lines of regression of xon y and y on x are respectively x 19.13 - 0.87y and y 11.64 - 0.50x. Find a The mean of x- series b The mean of y- series c The correlation coefficient between x and y. Sol. Let the mean of x-series is x and that of y-series be y. Since the lines of regression pass through x y we have x 19.13 - 0.87 y or - 0.87 y 19.13 . 1 and y 11.64 - 0.50x or 0.50x y 11.64 . 2 On solving 1 and 2 we get x 15.94 and y 3.67. Therefore mean of x-series 15.94 And mean of y-series 3.67 Now the line of regression of y on x is y 11.64 - 0.50x bx -0.50 yx Also the line of regresson x on y is x 19.13 - 0.87y bv -0.87 r byxbxy J - 0.50 - 0.87 f0.435 -0.66 Clearly r is taken as negative since each one of byx and bxy is negative. Example 10. Out of the following two regression lines find the line of regression of x ony 2x 3y 7 and 5x 4y 9. CURVE FITTING 417 Sol. Let 2x 3y 7 be the regression line of x on y. Then 5x 4y 9 is the regression line of y on x. Therefore 2x 3y 7 and 5x 4y 9 x -3y 7 and y -5 x 9 2 2 4 4 _ 3 Y 5 A r ibxibyx -a I I - l x yx Ail 2 II 4 3 5 bxy -2 and byx -4 v r bxy byx have the same sign 15 . -J -1 which is impossible. Therefore our choice of regression line is incorrect. Hence the regression line of x on y is 5x 4y 9. Ans. 2x Example 11. Find the correlation coefficient between x and y when the lines of regression are 9y 6 0 and x - 2y 1 0. Sol. Let the line of regression of x on y be 2x - 9y 6 0 Then the line of regression of y on x is x - 2y 1 0 . Therefore 2x - 9y 6 0 and x - 2y 1 0 9 x -y-3 and 1 1 y -x - 22 9 xy 2 and Yv 1 yx 2 r l b n xy yx ir 9 i a i -x- 12 2 1 which is impossible. 3 2 So our choice of regression line is incorrect. .