l Simple linear regression model Probabilistic Model: yi=B0+ Bx;+ a where yi=a value of the dependent variable, y x i=a value of the independent variable, x Bo= the y-intercept of the regression line Bi=the slope of the regression line 8= random error the residual Deterministic model: botb where and ii is the predicted value of y in contrast to the actual value of 1/ o 2002 The Wadsworth Group
Simple Linear Regression Model • Probabilistic Model: yi = b0 + b1xi + ei where yi = a value of the dependent variable, y xi = a value of the independent variable, x b0 = the y-intercept of the regression line b1 = the slope of the regression line ei = random error, the residual • Deterministic Model: = b0 + b1xi where and is the predicted value of y in contrast to the actual value of y. y ? i b 0 b 0 , b 1 b 1 y ? i © 2002 The Wadsworth Group
l Determining the least squares Regression line Least squares regression lines b. b Slope ∑xy;)-n衩 (∑x2)-n y-intercept bo =y? 6, x o 2002 The Wadsworth Group
Determining the Least Squares Regression Line • Least Squares Regression Line: – Slope – y-intercept y ˆ = b 0 + b 1 x 1 b 1 = ( x i y i ) – n×x ×y ( x i 2) – n×x 2 b 0 = y ? b 1 x © 2002 The Wadsworth Group
l Simple linear regression An example Problem 15.9: For a sample of 8 employees a personnel director has collected the following data on ownership of company stock, 1, versus years with the firm, x 61214 91315 y300408560252288650630522 (a) determine the least squares regression line and interpret its slope.(b)For an employee who has been with the firm 10 years, what is the predicted number of shares of stock owned? o 2002 The Wadsworth Group
Simple Linear Regression: An Example • Problem 15.9: For a sample of 8 employees, a personnel director has collected the following data on ownership of company stock, y, versus years with the firm, x. x 6 12 14 6 9 13 15 9 y 300 408 560 252 288 650 630 522 (a) Determine the least squares regression line and interpret its slope. (b) For an employee who has been with the firm 10 years, what is the predicted number of shares of stock owned? © 2002 The Wadsworth Group
Ⅷ An example,,cont. 6300 1800 36 12408 4896 144 14560 7840 196 6252 1512 36 9288 2592 81 13650 8450 169 15630 9450 225 9522 4698 81 Mean:10.545125 Sum. 41,238 968 o 2002 The Wadsworth Group
An Example, cont. x y x•y x 2 6 300 1800 36 12 408 4896 144 14 560 7840 196 6 252 1512 36 9 288 2592 81 13 650 8450 169 15 630 9450 225 9 522 4698 81 Mean: 10.5 451.25 Sum: 41,238 968 © 2002 The Wadsworth Group
Ⅷ An example,,cont. Slope (∑xy1)-n米41238-80.5)45125 38.7558 2)-nX2 968-810.5 °y- ntercept: y?b1x=45125?(38758)(10.5)=44.3140 So the best-fit linear model, rounding to the nearest tenth is j=44.3140+38.7558x≈44.3+38.8x o 2002 The Wadsworth Group
An Example, cont. • Slope: • y-Intercept: So the “best-fit” linear model, rounding to the nearest tenth, is: b 1 = ( x i y i ) – n×x ×y ( x i 2) – n×x 2 = 41238 – 8×(10.5)×(451.25) 968 - 8×(10.5) 2 = 38.7558 b 0 = y ? b 1 x = 451.25 ? (38.7558)(10.5) = 44.3140 y ˆ = 44.3140 + 38.7558x 44.3 + 38.8x © 2002 The Wadsworth Group