CHAPTER 15 Simple linear regression and correlation to accompany Introduction to business statistics fourth edition by ronald M. Weiers Presentation by Priscilla Chaffe-Stengel Donald n tenge o 2002 The Wadsworth Group
CHAPTER 15 Simple Linear Regression and Correlation to accompany Introduction to Business Statistics fourth edition, by Ronald M. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. Stengel © 2002 The Wadsworth Group
l Chapter 15-Learning objectives Determine the least squares regression equation, and make point and interval estimates for the dependent variable Determine and interpret the value of the Coefficient of correlation Coefficient of determination Construct confidence intervals and carry out hypothesis tests involving the slope of the regression line o 2002 The Wadsworth Group
Chapter 15 - Learning Objectives • Determine the least squares regression equation, and make point and interval estimates for the dependent variable. • Determine and interpret the value of the: – Coefficient of correlation. – Coefficient of determination. • Construct confidence intervals and carry out hypothesis tests involving the slope of the regression line. © 2002 The Wadsworth Group
l Chapter 15-Key Terms · Direct or inverse Confidence interval relationships or the mean Least squares Prediction interval for regression mode an individual value Standard error of the Coefficient of estimate, s correlation ° Point estimate using· Coefficient of the regression model determination o 2002 The Wadsworth Group
Chapter 15 - Key Terms • Direct or inverse relationships • Least squares regression model • Standard error of the estimate, sy,x • Point estimate using the regression model • Confidence interval for the mean • Prediction interval for an individual value • Coefficient of correlation • Coefficient of determination © 2002 The Wadsworth Group
l Chapter 15-Key concep Regression analysis generates a best-fit equation that can be used in predicting the values of the dependent variable as a function of the independent variable o 2002 The Wadsworth Group
Chapter 15 - Key Concept Regression analysis generates a “best-fit” mathematical equation that can be used in predicting the values of the dependent variable as a function of the independent variable. © 2002 The Wadsworth Group
l Direct us Inverse relationships Direct relationship As x increases, y increases The graph of the model rises from left to right The slope of the linear model is positive Inverse relationship As x increases, y decreases The graph of the model falls from left to right The slope of the linear model is negative o 2002 The Wadsworth Group
Direct vs Inverse Relationships • Direct relationship: – As x increases, y increases. – The graph of the model rises from left to right. – The slope of the linear model is positive. • Inverse relationship: – As x increases, y decreases. – The graph of the model falls from left to right. – The slope of the linear model is negative. © 2002 The Wadsworth Group
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
