PPT5 Simple Linear Regression0

PPT5 Simple Linear Regression0 - McGill University Advanced...

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McGill University Advanced Business Statistics MGSC-272
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Regression Simple Linear Regression
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Simple Linear Regression Simple linear regression analysis is used to analyze the nature of the relationship between two variables. The dependent variable is designated by Y and the independent variable is designated by X. For a given independent variable there is not just one possible value for the dependent variable. The decision regarding which variable to designate Y and which variable to designate X must be based upon theory, knowledge of the subject matter and the objectives of the analysis. The relationship between the two variables is estimated and then used to make predictions for Y.
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EXAMPLE The VP of Sales for a large department store chain wishes to investigate the relationship between store profits Y per day (in $1,000) and advertising expenditures per month ( X in $1,000). The following data has been determined from a random sample of 10 stores: Store Profit (Y) Advertising (X) 1 9.4 3 2 10.3 3 3 10.9 4 4 9.9 4 5 12.9 5 6 11.8 5 7 11.5 6 8 13.2 6 9 12.8 7 10 12.1 7
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Scatter Diagram A scatter diagram is a graph showing the shape and direction of the underlying relationship between the independent variable X and the dependent variable Y . Observations are plotted in pairs with one variable plotted on each axis.
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Scatter Diagram Profit vs Advertising Scatter Diagram 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 7 8 Advertising Profit
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Linear Relationships Between Two Variables Intercept and Slope The relationship between the 2 variables is described by a straight line model in the general form: True Regression Line: Y = β 0 + β 1 X + ε . Estimated Regression Line: = b 0 + b 1 X. Y
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Definitions of Terms X = the independent variable. β 0 = the true Y-intercept. β 1 = the true slope. ε = the error term. b 0 = the estimated Y -intercept (also denoted by ). b 1 = the estimated slope (also denoted by ). 0 ˆ β 1 ˆ β
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Notation 0 0 1 1 ˆ ˆ b b β β = =
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Simple Linear Regression Model Y i = β 0 + β 1 X i + ε i Assumptions: 1. Linearity. 2. For each value of X the Y values are normally distributed. 3. For each value of X the variance of the Y-values is the same (homoscedasticity). 4. Independence X i X j X k Y
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Simple Linear Regression Model Y i = β 0 + β 1 X i + ε i Y E(Y) = β 0 + β 1 X = b 0 +b 1 X ˆ Y X True Regression Line Estimated Regression Line
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Intercept and Slope The Y-intercept is the point on the Y-axis where the regression line crosses and is the average value of Y when X = 0. The regression line either slopes upward (positive slope) or downward (negative slope) and the slope represents the average change in Y when X is increased by one unit. The Y-intercept and slope are called the parameters of the regression line.
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The Least Squares Regression Line Minimizing the Error Sum of Squares Obviously there can be many lines that could be seen as fitting the data in a scatter diagram.
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