Regression ModelsLearning ObjectivesSlide 3Chapter OutlineSlide 5IntroductionSlide 7Scatter DiagramTriple A ConstructionSlide 10Simple Linear RegressionSlide 12Slide 13Least Squares RegressionSlide 15Slide 16Slide 17Slide 18Measuring the Fit of the Regression ModelSlide 20Slide 21Slide 22Slide 23Coefficient of DeterminationCorrelation CoefficientSlide 26Using Computer Software for RegressionSlide 28Slide 29Slide 30Slide 31Assumptions of the Regression ModelResidual PlotsSlide 34Slide 35Estimating the VarianceSlide 37Testing the Model for SignificanceSlide 39Slide 40Slide 41Steps in a Hypothesis TestSlide 43Slide 44Slide 45Slide 46r2 coefficient of determinationCoefficient HypothesesAnalysis of Variance (ANOVA) TableANOVA for Triple A ConstructionMultiple Regression AnalysisSlide 52Jenny Wilson RealtySlide 54Slide 55Evaluating Multiple Regression ModelsSlide 57Slide 58Binary or Dummy VariablesSlide 60Slide 61Slide 62Model BuildingSlide 64Slide 65Slide 66Nonlinear RegressionColonel MotorsSlide 69Slide 70Slide 71Slide 72Slide 73Cautions and PitfallsSlide 75© 2008 Prentice-Hall, Inc.Chapter 4To accompanyQuantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna Power Point slides created by Jeff HeylRegression Models© 2009 Prentice-Hall, Inc.© 2009 Prentice-Hall, Inc. 4 – 2Learning Objectives1. Identify variables and use them in a regression model2. Develop simple linear regression equations from sample data and interpret the slope and intercept3. Compute the coefficient of determination and the coefficient of correlation and interpret their meanings4. Interpret the F-test in a linear regression model5. List the assumptions used in regression and use residual plots to identify problemsAfter completing this chapter, students will be able to:After completing this chapter, students will be able to:© 2009 Prentice-Hall, Inc. 4 – 3Learning Objectives6. Develop a multiple regression model and use it to predict7. Use dummy variables to model categorical data8. Determine which variables should be included in a multiple regression model9. Transform a nonlinear function into a linear one for use in regression10. Understand and avoid common mistakes made in the use of regression analysisAfter completing this chapter, students will be able to:After completing this chapter, students will be able to:© 2009 Prentice-Hall, Inc. 4 – 4Chapter Outline4.1 Introduction4.2 Scatter Diagrams4.3 Simple Linear Regression4.4 Measuring the Fit of the Regression Model4.5 Using Computer Software for Regression4.6 Assumptions of the Regression Model© 2009 Prentice-Hall, Inc. 4 – 5Chapter Outline4.7 Testing the Model for Significance4.8 Multiple Regression Analysis4.9 Binary or Dummy Variables4.10 Model Building4.11 Nonlinear Regression 4.12 Cautions and Pitfalls in Regression Analysis© 2009 Prentice-Hall, Inc. 4 – 6IntroductionRegression analysisRegression analysis is a very valuable tool for a managerRegression can be used toUnderstand the relationship between variablesPredict the value of one variable based on another variableExamplesDetermining best location for a new storeStudying the effectiveness of advertising dollars in increasing sales volume© 2009 Prentice-Hall, Inc. 4 – 7IntroductionThe variable to be predicted is called the dependent variabledependent variable Sometimes called the response variableresponse variableThe value of this variable depends on the value of the independent variableindependent variableSometimes called the explanatoryexplanatory or predictor variablepredictor variableIndependent variableDependent variableIndependent variable= +© 2009 Prentice-Hall, Inc. 4 – 8Scatter DiagramGraphing is a helpful way to investigate the relationship between variablesA scatter diagramscatter diagram or scatter plotscatter plot is often used The independent variable is normally plotted on the X axisThe dependent variable is normally plotted on the Y axis© 2009 Prentice-Hall, Inc. 4 – 9Triple A ConstructionTriple A Construction renovates old homesThey have found that the dollar volume of renovation work is dependent on the area payrollTRIPLE A’S SALES($100,000’s)LOCAL PAYROLL($100,000,000’s)6 38 49 65 44.5 29.5 5Table 4.1© 2009 Prentice-Hall, Inc. 4 – 10Triple A ConstructionFigure 4.112 –10 –8 –6 –4 –2 –0 –Sales ($100,000)Payroll ($100 million)| | | | | | | |0 1 2 3 4 5 6 7 8© 2009 Prentice-Hall, Inc. 4 – 11Simple Linear RegressionwhereY = dependent variable (response) X = independent variable (predictor or explanatory) 0= intercept (value of Y when X = 0) 1= slope of the regression line = random errorRegression models are used to test if there is a relationship between variables (predict sales based on payroll)There is some random error that cannot be predicted XY10© 2009 Prentice-Hall, Inc. 4 – 12Simple Linear RegressionTrue values for the slope and intercept are not known so they are estimated using sample dataXbbY10ˆwhere Y = dependent variable (response) X = independent variable (predictor or explanatory) b0= intercept (value of Y when X = 0) b1= slope of the regression line^© 2009 Prentice-Hall, Inc. 4 – 13Triple A ConstructionTriple A Construction is trying to predict sales based on area payrollY = SalesX = Area payrollThe line chosen in Figure 4.1 is the one that minimizes the errorsError = (Actual value) – (Predicted value)YYeˆ© 2009 Prentice-Hall, Inc. 4 – 14Least Squares RegressionErrors can be positive or negative so the average error could be zero even though individual errors could be large. Least squares regression minimizes the sum of the squared errors.© 2009 Prentice-Hall, Inc. 4 – 15Triple A ConstructionFor the simple linear regression model, the values of the intercept and slope can be calculated using the formulas belowXbbY10ˆvalues of (mean) average XnXX values of (mean) average YnYY 21)())((XXYYXXbXbYb10© 2009 Prentice-Hall, Inc. 4 – 16Triple A ConstructionY X(X – X)2(X – X)(Y – Y)6 3 (3 – 4)2 = 1 (3 – 4)(6 – 7) = 18 4 (4 – 4)2 = 0 (4 – 4)(8 – 7) = 09 6 (6 – 4)2 = 4 (6 – 4)(9 – 7) = 45 4 (4 – 4)2 = 0 (4 – 4)(5 – 7) = 04.5 2 (2 – 4)2 = 4 (2 – 4)(4.5 – 7) = 59.5 5 (5 – 4)2 = 1 (5 – 4)(9.5 – 7) = 2.5ΣY = 42Y = 42/6 = 7ΣX = 24X =