# SLR - Simple Linear Regression Simple Linear Regression...

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Simple Linear Regression

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BUAD 310 2 Simple Linear Regression Simple Linear Regression Model Least Squares Point Estimates Model Assumptions Variation Coefficient of Determination Inference F-Test Coefficient t-tests CIs and Prediction Intervals (PI)
BUAD 310 3 y|x = + X is mean value of y when value of X is given. is Y-intercept , mean of Y when X is 0. * is slope , change in mean of y per unit change in X . e is error term describing leftover effect on Y . *Note: be careful: You need to have data where X is 0 for this to make sense. Simple Linear Regression Model Average Hourly Weekly Fuel Temperature Consumption Week x (deg F) y (MMcf) 1 28.0 12.4 2 28.0 11.7 3 32.5 12.4 4 39.0 10.8 5 45.9 9.4 6 57.8 9.5 7 58.1 8.0 8 62.5 7.5 Y = y|x  e = 0  1 x  e MINITAB: Graph Scatterplot

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BUAD 310 4 Simple Linear Regression Model LS minimizes squared error
BUAD 310 5 Slope (b 1 ) Y-Intercept (b 0 ) Y X 12.4 28.0 11.7 28.0 12.4 32.5 10.8 39.0 9.4 45.9 9.5 57.8 8.0 58.1 7.5 62.5 81.7 351.8 Prediction (X = 40) Least Squares Point Estimate ˆ Y b 0 b 1 X 15.84 0.1279 40   10.72 b 1 X i X Y i Y X i X 2 r XY s Y s X -179.6475 1404.355 -0.1279 Y y i n 81.7 8 10.2125, X x i n 351.8 8 43.98 b 0 Y - b 1 X 10.2125-(-0.1279)(43.98) 15.84

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BUAD 310 6 Assumptions about the model error terms 1. Normality Error terms follow a normal distribution for all values of x. 2. Constant Variance Variance of error terms s 2 is the same for all values of X. 3. Independence Values of error terms are statistically independent of each other. 4. Linearity Linear in parameters. Regression Model Assumptions
BUAD 310 7 Regression Model Assumptions

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BUAD 310 8 Variation of actual Y from predicted Y Measured by standard error of estimate Sample standard deviation of e Denoted by “ s e Affects several factors Parameter significance Prediction accuracy Random Error Variation
BUAD 310 9     2 2 2 ˆ - - 2 - 2 2 ii e YY e e e s n n n  • Measures standard deviation of predicted vs. actual values •Would get the same number if saved error terms and calculated the standard deviation of them •Measures average error of estimate Standard Error of Estimate

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