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4.10_linear_reg

# 4.10_linear_reg - Linear Regression 25 20 15 y 10 5 0-5 0 5...

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1 Mike Pore - Sept 2004 Linear Regression 0 5 10 15 20 25 y -5 0 5 10 15 20 25 30 35 x Linear Fit

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2 Mike Pore - Sept 2004 Topics Scatterplots Correlation Least squares regression Residual plots Causation Tests of regression Transformations Probability intervals
3 Mike Pore - Sept 2004 Scatterplots In the beginning, we have paired variables: (x 1 , y 1 ), (x 2 , y 2 ), . . . , (x n , y n ) The Y are observations The X can be either observations or observer determined values of the X variable.

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4 Mike Pore - Sept 2004 Scatterplots 0 5 10 15 20 25 y -5 0 5 10 15 20 25 30 35 x
5 Mike Pore - Sept 2004 Modeling the Data with a Line 0 5 10 15 20 25 y -5 0 5 10 15 20 25 30 35 x Linear Fit

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6 Mike Pore - Sept 2004 Graphing a Line x y b x b b y = + = 1 1 0 x 1 x 2 y 1 y 2 (x 1 , y 1 ) (x 2 , y 2 ) (0, b 0 ) x y
7 Mike Pore - Sept 2004 Graphing a Line The slope: b 1 0 5 10 15 20 x Positive: up to the right Y=x has slope 1 Slope 2 is half way from 1 to vertical Slope 4 is half way from 2 to vertical Slope ½ is half way from 1 to horizonal Slope 1/4 is half way from ½ to horizonal Negative: down to the right Y=-x has slope -1 Slope -2 is half way from -1 to vertical Slope -4 is half way from -2 to vertical Slope -½ is half way from -1 to horizonal Slope -1/4 is half way from -½ to horizonal Zero: horizonal

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8 Mike Pore - Sept 2004 Data Calculations i x i y i 1 2 n ( 29 x x i - ( 29 y y i - ( 29 2 x x i - ( 29 2 y y i - ( 29 ( 29 y y x x i i - - ( 29 ( 29 - - = - - = 2 2 1 1 1 1 y y n s x x n s i y i x - - - = y i x i s y y s x x n r n correlatio 1 1
9 Mike Pore - Sept 2004 Correlation

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