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5 properties of regression co efficient 1 both

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Unformatted text preview: 19.5 5a = 59.5 59.5 a= = 11.9 5 Hence, required regression line Y on X is Y = 11.9 – 0.65 X. Again, regression equation of X on Y is X = a + bY and or b = - The normal equations are ∑X = na + b∑Y and ∑XY = a∑Y + b∑Y2 223 Now, substituting the corresponding values from the above table, we get 30 = 5a + 40b …(3) . 214 = 40a + 340b …(4) . Multiplying (3) by 8, we get 240 = 40a + 320 b …(5) . (4) – (5) gives -26 = 20b 26 b== - 1.3 20 Substituting b = - 1.3 in equation (3) gives 30 = 5a – 52 5a = 82 82 a= = 16.4 5 Hence, Required regression line of X on Y is X = 16.4 – 1.3Y (ii) Regression Co-efficents: σ The regression equation of Y on X is ye = y + r y ( x − x) σx Here, the regression Co.efficient of Y on X is σ b1 = byx = r y σx ye = y + b1 ( x − x) The regression equation of X on Y is σ X e = x + r x ( y − y) σy Here, the regression Co-efficient of X on Y σ b2 = bxy = r x σy X e = X + b2 ( y − y ) 224 If the deviation are taken from respective means of x and y xy ( X − X )(Y − Y ) b1 = byx = ∑ and = ∑2 x ( X − X )2 ∑ ∑ b2 = bxy = ∑ ( X − X )(Y − Y ) ∑ (Y − Y ) 2 =∑ xy ∑y 2 where x = X − X , y = Y − Y If the deviations are taken from any arbitrary values of x and y (short – cut method) n uv − ∑ u ∑ v b1 = byx = ∑ 2 n∑ u 2 − ( ∑ u ) b2 = bxy = n ∑ uv − ∑ u ∑ v n∑ v 2 − ( ∑ v ) 2 where u = x – A : v = Y-B A = any value in X B = any value in Y 9.5 Properties of Regression Co-efficient: 1. Both regression coefficients must have the same sign, ie either they will be positive or negative. 2. correlation coefficient is the geometric mean of the regression coefficients ie, r = ± b1b2 3. The correlation coefficient will have the same sign as that of the regression coefficients. 4. If one regression coefficient is greater than unity, then other regression coefficient must be less than unity. 5. Regression coefficients are independent of origin but not of scale. 6. Arithmetic mean of b1 and b2 is equal to or greater than the b1 + b2 coeffic...
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This note was uploaded on 01/18/2014 for the course BUS 100 taught by Professor Moshiri during the Winter '08 term at UC Riverside.

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