ch_14_Notes - Chapter 14 Simple Linear Regression Outline:...

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Chapter 14 Simple Linear Regression Outline: Simple Linear Regression Model Least Squares Method Coefficient of Determination Model Assumptions Testing for Significance Using the Estimated Regression Equation for Estimation and Prediction Residual Analysis: Validating Model Assumptions What is regression for? Describe the relationship between a response/dependent variable and at least one exploratory/independent variable. Used for prediction e.g., Sales versus promotion activities Correlation Analysis Measures the association of numerical values e.g., Euro and U.S. dollars 1
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Steps of Regression Modeling: Define problem Specify model: for example, y = βx + ε Collect Data: ((x 1 , y 1 ), (x 2 , y 2 ), … (x n , y n )) Descriptive Data Analysis Estimate unknown parameters Evaluate Model Use Model for prediction Step 1: Define Problem What are the model objectives? Who will use the model? What are the benefits? Are resources available (data)? How will the results be implemented? Example: Develop a model to explain the variations in sales by advertising and promotional activities. 2
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Step 2: Specify Model-- Which is logical? Simple Linear Regression Models Widely used for trends analysis One dependent variable ( i y is the dependent variable) One independent variable ( i x is independent/exploratory variable) 3 Advertising Sale s Advertising Sales Advertising Sales Advertising Sales Y Y  =  mX  +  b X m  = Slope b   Y -intercept
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Three possible regression lines in simple linear regression (Figure 14.1 ASW). 4
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Step 3: Collect Data Step 4: Descriptive Data Analysis Scatter Diagram Correlation Coefficient Example: 5 Population $ $ $ $ Unknown  Relationship ε β + + = X Y 1 0 Random Sample $ $ $ $ e X b b Y + + = 1 0 promotion versus sales 80 85 90 95 100 105 110 115 120 125 80 85 90 95 100 105 110 115 120 promotion sales
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Correlation Coefficients: Linear association between variables The Simple Linear Regression Model: Simple Linear Regression Model y = β 0 + 1 x + ε (randomness) Simple Linear Regression Equation E( y ) = 0 + 1 x Estimated Simple Linear Regression Equation y ˆ = b 0 + b 1 x (propagated randomness) The estimated process in simple linear regression (Figure 14.2 ASW). Y X r  = 0 Y X r  = 0 Y X r  = -1 Y X r  = 1 6
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Step 5: Estimate Unknown Parameters Least Squares Method: Error or Residual: i i i Y Y e ˆ - = Least Squares Criterion: Minimize the sum of squared errors. min = - n i i i y y 1 2 ) ˆ ( = min = n i i e 1 2 ) ( where: i y = observed value of the dependent variable for the i th observation i y ˆ = estimated value of the dependent variable for the i th observation Why squares? e 2 e 1 e 3 e 4 Y X i i X b b Y 1 0 ˆ + = 7
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Least Square Estimators: Sum of squares and cross products: Regression estimators: Slope for the Estimated Regression Equation y -Intercept for the Estimated Regression Equation x b y b 1 0 - = where: Y X 8 - - = - - - = n x x n y x y x x x y y x x b i i i i i i i i i / ) ( / ) ( ) ( ) )( ( 2 2 2 1 ( 29 ( 29 ( 29 ( 29 n y
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This note was uploaded on 01/31/2011 for the course MGMT 305 taught by Professor Priya during the Spring '08 term at Purdue University-West Lafayette.

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ch_14_Notes - Chapter 14 Simple Linear Regression Outline:...

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