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Simple%20Linear%20Regression-1

Simple%20Linear%20Regression-1 - andCorrelation...

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Click to edit Master subtitle style 4/24/11 Simple Linear Regression and Correlation

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4/24/11 Introduction Regression analysis generates a “best-fit”  mathematical equation that can be used in  predicting  the values of the dependent  variable as a function of the independent  variable.
4/24/11 Direct vs Inverse Relationships Direct  relationship: As  x  increases,  y  increases. The graph of the model rises from left to right. The slope of the linear model is positive. Inverse  relationship: As  x  increases,  y  decreases. The graph of the model falls from left to right. The slope of the linear model is negative.

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4/24/11 Direct vs Inverse Relationships 5 6 7 8 9 10 11 200 220 240 260 280 300 320 340 360 380 400     x
4/24/11 Simple Linear Regression Model Probabilistic Model: yi  =  β 0 +   1 xi  + ε i       where yi  = a value of the dependent variable,  y xi  = a value of the independent variable,  x 0 = the  y -intercept of the regression line 1 = the slope of the regression line i  = random error, the residual Deterministic Model:  =  b 0 +  b 1 xi where and     is the predicted value of  in contrast to the  actual value of  y . ˆ y i b 0 ≈β 0 , b 1 1 ˆ y i

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4/24/11 Simple Linear Regression Model Assumptions: linearity  of the relationship between  dependent and independent variables  independence  of the errors constant variance  of the errors  versus time  versus the predictions (or versus any  independent variable)  normality  of the error distribution.
4/24/11 Determining the Least Squares Regression Line Least Squares Regression Line: Slope y -intercept ˆ y = b + b x = ( x i y i ) x y ( x i 2 x b 0 = y b 1 x

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4/24/11 Simple Linear Regression: An Example Problem 15.9:  For a sample of 8 employees, a personnel director has  collected the following data on ownership of company  stock,  y , versus years with the firm,  x . x     6   12   14     6     9   13   15     9 y 300 408 560 252 288 650 630 522 (a) Determine the least squares regression line and
4/24/11 Problem 15.9 5 6 7 8 9 10 11 200 220 240 260 280 300 320 340 360 380 400   Linear Regression for

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4/24/11 Problem 15.9, cont.    x     y      x•y      x   6 300 1800   36 12 408 4896 144 14 560 7840 196   6 252 1512   36   9 288 2592   81 13 650 8450 169 15 630 9450 225   9 522 4698   81 Mean:10.5 451.25
4/24/11 Problem 15.9, cont.

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