Chapter14 Multiple Regression
CQMS202-Business Statistics II
Chapter14
Chapter14 Multiple Regression Outcomes: 1. Describe the relationship between several independent variables and a dependent variable using a multiple regression 2. Compute and interpret the multiple standard error and the coefficient of determination 3. Conduct a Global test (F test) to determine the usefulness of the model 4. Conduct a test of hypothesis on each of the regression coefficient. 5. Use the multiple regression model for estimation and prediction
Multiple Regression Models 1. The simple linear regression model was used to analyze how one quantitative variable (the dependent variable y) is related to one other quantitative variable (the independent variable x). 2. Multiple regression allows for any number of independent variables.
Example 1 The director of marketing at Ryerson Wholesale Products is studying the monthly sales. Three independent variables were selected as estimators of sales: regional population ( x1 ) , per-capita income ( x 2 ) and regional unemployment rate ( x 3 ). The regression equation was computed to be (in dollars): yˆ = 64100 + 0.394 x1 +9.6 x 2 −11600 x3
a. What is the full name of the equation? b. Interpret the number 64100 c. What are the estimated monthly sales for a particular region with a population of 796000 per-capita income of $6940, and an Page#1
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Chapter14
unemployment rate of 6.0 percent?(Use 6.0 in your calculation)
3. We expect to develop models that fit the data better than would a simple linear regression model.
4. General form of the Multiple Regression Model y = β0 + β1 x1 + β2 x 2 + ... + βk x k + ε
where y is the dependent variable x1 , x 2 ,..., x k are independent variables E ( y ) = β0 + β1 x1 + β2 x 2 + ... + βk x k is the deterministic portion of the model βi determines the contribution of the independent variable xi
Note: The symbols x1 , x 2 ,..., x k may represent higher-order terms. For example, x1 might represent the current interest rate, x 2 might represent x12 , and so forth.
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Example2 The following are data on horsepower ( x1 ) , time from zero to 60 miles per hour ( x 2 ) , top speed ( x 3 ) , miles per gallon ( x 4 ) , and price ( y ) in thousands of dollars for 10 sports cars. (Road & Track, October 1998) BMW M3 Corvette Dodge Viper Ford Mustang Honda Prelude Mitsubishi GT Toyota Supra Nissan 300ZX Alfa Romeo Mazda RX-7
x1
x2
x3
x4
y
240 300 400 240 190 320 320 300 320 255
6.0 5.7 4.8 6.9 7.1 5.7 5.3 6.0 7.6 5.5
120 170 160 140 139 159 155 155 150 158
24.6 16.8 14.0 18.0 24.0 16.3 18.8 18.7 17.5 17.0
38.4 41.4 54.8 25.8 25.6 43.7 48.2 40.8 38.1 35.0
a. Develop an estimated regression equation with horsepower, time from zero to 60 miles per hour, top speed, and miles per gallon as the four independent variables to predict price. b. For part (a) conduct a test to determine if the model appears useful. At 0.05 level of significance, what is your conclusion? c. For part (a) use the t test to determine the significance of each independent variable. At the 0.05 level of significance, what is your conclusion? d. Which of the four given models is most efficient? Why?
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e. Repeat part (a) with miles per gallon (mpg) removed from the list of independent variables. What is the new estimated regression equation? What is the coefficient of determination?
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Chapter14
Chapter14 Multiple Regression Outcomes: 1. Describe the relationship between several independent variables and a dependent variable using a multiple regression 2. Compute and interpret the multiple standard error and the coefficient of determination 3. Conduct a Global test (F test) to determine the usefulness of the model 4. Conduct a test of hypothesis on each of the regression coefficient. 5. Use the multiple regression model for estimation and prediction
Multiple Regression Models 1. The simple linear regression model was used to analyze how one quantitative variable (the dependent variable y) is related to one other quantitative variable (the independent variable x). 2. Multiple regression allows for any number of independent variables.
Example 1 The director of marketing at Ryerson Wholesale Products is studying the monthly sales. Three independent variables were selected as estimators of sales: regional population ( x1 ) , per-capita income ( x 2 ) and regional unemployment rate ( x 3 ). The regression equation was computed to be (in dollars): yˆ = 64100 + 0.394 x1 +9.6 x 2 −11600 x3
a. What is the full name of the equation? b. Interpret the number 64100 c. What are the estimated monthly sales for a particular region with a population of 796000 per-capita income of $6940, and an Page#1
Fall2010
www.notesolution.com
CQMS202-Business Statistics II
Chapter14
unemployment rate of 6.0 percent?(Use 6.0 in your calculation)
3. We expect to develop models that fit the data better than would a simple linear regression model.
4. General form of the Multiple Regression Model y = β0 + β1 x1 + β2 x 2 + ... + βk x k + ε
where y is the dependent variable x1 , x 2 ,..., x k are independent variables E ( y ) = β0 + β1 x1 + β2 x 2 + ... + βk x k is the deterministic portion of the model βi determines the contribution of the independent variable xi
Note: The symbols x1 , x 2 ,..., x k may represent higher-order terms. For example, x1 might represent the current interest rate, x 2 might represent x12 , and so forth.
Page#2
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Chapter14
Example2 The following are data on horsepower ( x1 ) , time from zero to 60 miles per hour ( x 2 ) , top speed ( x 3 ) , miles per gallon ( x 4 ) , and price ( y ) in thousands of dollars for 10 sports cars. (Road & Track, October 1998) BMW M3 Corvette Dodge Viper Ford Mustang Honda Prelude Mitsubishi GT Toyota Supra Nissan 300ZX Alfa Romeo Mazda RX-7
x1
x2
x3
x4
y
240 300 400 240 190 320 320 300 320 255
6.0 5.7 4.8 6.9 7.1 5.7 5.3 6.0 7.6 5.5
120 170 160 140 139 159 155 155 150 158
24.6 16.8 14.0 18.0 24.0 16.3 18.8 18.7 17.5 17.0
38.4 41.4 54.8 25.8 25.6 43.7 48.2 40.8 38.1 35.0
a. Develop an estimated regression equation with horsepower, time from zero to 60 miles per hour, top speed, and miles per gallon as the four independent variables to predict price. b. For part (a) conduct a test to determine if the model appears useful. At 0.05 level of significance, what is your conclusion? c. For part (a) use the t test to determine the significance of each independent variable. At the 0.05 level of significance, what is your conclusion? d. Which of the four given models is most efficient? Why?
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e. Repeat part (a) with miles per gallon (mpg) removed from the list of independent variables. What is the new estimated regression equation? What is the coefficient of determination?
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