© 2000 Prentice-Hall, Inc. Chap Chapter 10 Multiple Regression Models Business Statistics A First Course (2nd Edition)

© 2000 Prentice-Hall, Inc. Chap Chapter Topics The multiple regression model Residual analysis Testing for the significance of the regression model Inferences on the population regression coefficients Testing portions of the multiple regression model

© 2000 Prentice-Hall, Inc. Chap Chapter Topics The curvilinear regression model Dummy variables Collinearity Model building Pitfalls in multiple regression and ethical issues

© 2000 Prentice-Hall, Inc. Chap The Multiple Regression Model Relationship between 1 dependent & 2 or more independent variables is a linear function Population Y-intercept Population slopes Dependent (Response) variable for sample Independent (Explanatory) variables for sample model Random Error

© 2000 Prentice-Hall, Inc. Chap Bivariate model Population Multiple Regression Model

© 2000 Prentice-Hall, Inc. Chap Bivariate model Sample Multiple Regression Model

© 2000 Prentice-Hall, Inc. Chap Too complicated by hand! Ouch! Multiple Linear Regression Equation

© 2000 Prentice-Hall, Inc. Chap Multiple Regression Model: Example ( 0 F) Develop a model for estimating heating oil used for a single family home in the month of January based on average temperature and amount of insulation in inches.

© 2000 Prentice-Hall, Inc. Chap Sample Multiple Regression Model: Example Excel Output For each degree increase in temperature, the estimated average amount of heating oil used is decreased by gallons, holding insulation constant. For each increase in one inch of insulation, the estimated average use of heating oil is decreased by gallons, holding temperature constant.

© 2000 Prentice-Hall, Inc. Chap Slope (b i )  Estimated average Y changes by b i for each 1 unit increase in X i holding all other variables constant (ceterus paribus) Example: If b 1 = -2, then fuel oil usage (Y) is expected to decrease by an estimated 2 gallons for each 1 degree increase in temperature (X 1 ) given the inches of insulation (X 2 ) Interpretation of Estimated Coefficients

© 2000 Prentice-Hall, Inc. Chap Y-Intercept (b 0 )  Estimated Average value of Y when all X i = 0 Interpretation of Estimated Coefficients (continued)

© 2000 Prentice-Hall, Inc. Chap Using The Model to Make Predictions Predict the amount of heating oil used for a home if the average temperature is 30 0 and the insulation is 6 inches. The predicted heating oil used is gallons

© 2000 Prentice-Hall, Inc. Chap Coefficient of Multiple Determination Excel Output Adjusted r 2 reflects the number of explanatory variables and sample size is smaller than r 2

© 2000 Prentice-Hall, Inc. Chap Residuals vs y i  May need to transform Y variable Residuals vs X 1  May need to transform x 1 variable Residuals vs X 2  May need to transform X 2 variable Residuals vs time  May have autocorrelation Residual Plots 

© 2000 Prentice-Hall, Inc. Chap Residual Plots: Example Excel Output No Discernable Pattern

© 2000 Prentice-Hall, Inc. Chap Testing for Overall Significance Shows if there is a linear relationship between all of the X variables together and Y Use F test Statistic Hypotheses: H 0 :  1 =  2 = … =  p = 0 (No linear relationship) H 1 : At least one  i  0 ( At least one independent variable affects Y) The Null Hypothesis Is a Very Strong Statement Almost Always End Up with Rejecting the Null

© 2000 Prentice-Hall, Inc. Chap Test for Overall Significance Excel Output: Example p = 2, the number of explanatory variables n - 1 MSR MSE p value = F Test Statistic

© 2000 Prentice-Hall, Inc. Chap F H 0 :  1 =  2 = … =  p = 0 H 1 : At least one  I  0  =.05 Df = 2 and 12 Critical value(s): Test Statistic: Decision: Conclusion: Reject at  = 0.05 There is evidence that At least one independent variable affects Y  = 0.05 F  Test for Overall Significance Example Solution (Excel Output)

© 2000 Prentice-Hall, Inc. Chap Test for Significance: Individual Variables Shows if there is a linear relationship between the variable X i and Y Use t test Statistic Hypotheses: H 0 :  i = 0 (No linear relationship) H 1 :  i  0 (Linear relationship between X i and Y)

© 2000 Prentice-Hall, Inc. Chap T Test Statistic Excel Output: Example t Test Statistic for X 1 (Temperature) t Test Statistic for X 2 (Insulation)

© 2000 Prentice-Hall, Inc. Chap H 0 :  1 = 0 h 1 :  1  0 Df = 12 critical value(s): Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence of a significant effect of temperature on oil consumption. Z Reject H Does temperature have a significant effect on monthly consumption of heating oil? Test at  = T Test : Example Solution t Test Statistic =