Chapter 16 Multiple 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 16 Multiple 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 16 Learning objectives Obtain and interpret the multiple regression equation Make estimates using the regression model point value of the dependent variable, y Intervals > Confidence interval for the conditional mean of Prediction interval for an individual y observation Conduct and interpret hypothesis tests on the Coefficient of multiple determination Partial regression coefficients o 2002 The Wadsworth Group
Chapter 16 Learning Objectives • Obtain and interpret the multiple regression equation • Make estimates using the regression model: – Point value of the dependent variable, y – Intervals: »Confidence interval for the conditional mean of y »Prediction interval for an individual y observation • Conduct and interpret hypothesis tests on the – Coefficient of multiple determination – Partial regression coefficients © 2002 The Wadsworth Group
l Chapter 16-Key Terms Partial regression coefficients Multiple standard error of the estimate Conditional mean of Individual y observation Coefficient of multiple determination Coefficient of partial determination Global f-test Standard deviation of b o 2002 The Wadsworth Group
Chapter 16 - Key Terms • Partial regression coefficients • Multiple standard error of the estimate • Conditional mean of y • Individual y observation • Coefficient of multiple determination • Coefficient of partial determination • Global F-test • Standard deviation of bi © 2002 The Wadsworth Group
l The Multiple regression model ● Probabilistic model Bo+β1x1+β2x2+…+Bkx+ where yi=a value of the dependent variable, y Bo= the y-intercept Mli air., ki= individual values of the independent variables, xl x2,.,x B1,B2灬…,Bk= the partial regression coefficients for the independent variables, x1x,.,k Ci= random error, the residual o 2002 The Wadsworth Group
The Multiple Regression Model • Probabilistic Model yi = b0 + b1x1i + b2x2i + ... + bkxki + ei where yi = a value of the dependent variable, y b0 = the y-intercept x1i , x2i , ... , xki = individual values of the independent variables, x1 , x2 , ... , xk b1 , b2 ,... , bk = the partial regression coefficients for the independent variables, x1 , x2 , ... , xk ei = random error, the residual © 2002 The Wadsworth Group
l The Multiple regression model Sample regression equation bo+b1x1+b2x2+…+bk where J, the predicted value of the dependent variable, y, given the values of xi x2,.,xk bo= the y-intercept 1讠2i xki= individual values of the independent variables, Iv x,,xp 1.., bk=the partial regression coefficients for the independent variables, x1 x,.,xk o 2002 The Wadsworth Group
The Multiple Regression Model • Sample Regression Equation = b0 + b1x1i + b2x2i + ... + bkxki where = the predicted value of the dependent variable, y, given the values of x1 , x2 , ... , xk b0 = the y-intercept x1i , x2i , ... , xki = individual values of the independent variables, x1 , x2 , ... , xk b1 , b2 , ... , bk = the partial regression coefficients for the independent variables, x1 , x2 , ... , xk y ? i y ? i © 2002 The Wadsworth Group
l The Amount of Scatter in the data The multiple standard error of the estimate (-)2 n-k-1 where 1; =each observed value of y in the data set i,=the value of y that would have been estimated from the regression equation n= the number of data values in the set k=the number of independent(a) variables measures the dispersion of the data points around the regression hyperplane o 2002 The Wadsworth Group
The Amount of Scatter in the Data • The multiple standard error of the estimate where yi = each observed value of y in the data set = the value of y that would have been estimated from the regression equation n = the number of data values in the set k = the number of independent (x) variables measures the dispersion of the data points around the regression hyperplane. s e = (y i –y ˆ i )2 n–k–1 y ? i © 2002 The Wadsworth Group
I Approximating a Confidence Interval for a Mean of y a reasonable estimate for interval bounds on the conditional mean of y given various x values is generated by 予士 where the estimated value of y based on the set of x values provided t= critical t value,(1-a)% confidence, df=n-k-1 s, the multiple standard error of the estimate o 2002 The Wadsworth Group
Approximating a Confidence Interval for a Mean of y • A reasonable estimate for interval bounds on the conditional mean of y given various x values is generated by: where = the estimated value of y based on the set of x values provided t = critical t value, (1–a)% confidence, df = n – k – 1 se = the multiple standard error of the estimate n e s y ˆ ±t× y ? © 2002 The Wadsworth Group
I Approximating a prediction Interval for an Individualy value a reasonable estimate for interval bounds on an individual y value given various x values is generated by t where i= the estimated value of y based on the set of x values provided t=critical t value, (1-0)% confidence df=n-k s,= the multiple standard error of the estimate o 2002 The Wadsworth Group
Approximating a Prediction Interval for an Individual y Value • A reasonable estimate for interval bounds on an individual y value given various x values is generated by: where = the estimated value of y based on the set of x values provided t = critical t value, (1–a)% confidence, df = n – k – 1 se = the multiple standard error of the estimate y ˆ ±t×s e y ? © 2002 The Wadsworth Group
l Coefficient of multiple Determination The proportion of variance in y that is explained by the multiple regression equation is given by ∑(y.-y R=1 21 SSE SSR ∑(y.-y SSTSST o 2002 The Wadsworth Group
Coefficient of Multiple Determination • The proportion of variance in y that is explained by the multiple regression equation is given by: R 2 = 1– S(y i –y ˆ i ) 2 S(y i –y ) 2 = 1 – SSE SST = SSR SST © 2002 The Wadsworth Group
l Coefficients of partial Determination For each independent variable, the coefficient of partial determination denotes the proportion of total variation in y that is explained by that one independent variable alone, holding the values of all other independent variables constant. The coefficients are reported on computer printouts o 2002 The Wadsworth Group
Coefficients of Partial Determination • For each independent variable, the coefficient of partial determination denotes the proportion of total variation in y that is explained by that one independent variable alone, holding the values of all other independent variables constant. The coefficients are reported on computer printouts. © 2002 The Wadsworth Group
