1. ESTIMATING THE REGRESSION COEFFICIENTS FOR SIMPLE LINEAR REGRESSION

2. Step 2: Estimating β 1 and β 0 In simple linear regression, in Step 1, it was hypothesized that: y =  0 +  1 x +   0  1In Step 2, the best estimates for  0 and  1 are determined. b 0 b 1These “best estimates” are designated as b 0 and b 1 respectively.

3. Notation

4. DETERMINING THE BEST STRAIGHT LINE The best straight line is the one that, in some sense, minimizes the overall errors. But the positive values for the errors will offset the negative values giving an average error value of 0. squaredTo make sure all quantities are positive -- the errors are squared. THE SUM OF THE SQUARED ERROR (SSE)THE BEST STRAIGHT LINE MINIMIZES THE SUM OF THE SQUARED ERROR (SSE)

5. MINIMIZING SSE We want to minimize SSE where: This is a function in two variables: b 0 and b 1.

6. Method of Least Squares METHOD OF LEAST SQUARES.Because we are minimizing the sum of the squared errors, the approach for doing this is called the METHOD OF LEAST SQUARES. To find the minimum of a function of two variables (b 0 and b 1 ), take partial derivatives with respect to each of the variables and set them equal to 0. NORMAL EQUATIONSWe then have two equations in the two unknowns and we can solve for the values of the two unknowns -- these are known as the NORMAL EQUATIONS for regression.

7. THE PARTIAL DERIVATIVES The result from taking the partial derivatives of SSE and setting them equal to 0 is: normal equations Simplifying gives the two normal equations for b 0 and b 1 :

8. SOLVING FOR b 1 THE BEST ESTIMATE FOR  1 Solving the normal equations for b 1 gives: Doing a little algebra, gives these three alternate formulas:

9. SOLVING FOR b 0 THE BEST ESTIMATE FOR β 0 Regardless of how b 1 is calculated, b 0 is found by: And the regression equation is:

10. Example – The Data 11200101000 280092000 31000110000 41300120000 570090000 680082000 7100093000 860075000 990091000 101100105000 SUM 9400 959000

11. Example – Table Calculations 112001010002605100132600067600 280092000-140-390054600019600 3100011000060141008460003600 41300120000360241008676000129600 570090000-240-5900141600057600 680082000-140-13900194600019600 710009300060-2900-1740003600 860075000-340-209007106000115600 990091000-40-49001960001600 1011001050001609100145600025600 SUM 23,340,000 444,000

12. CALCULATING b 1 AND b 0 THE REGRESSION EQUATION Thus the estimated regression equation is:

13. What does the model predict sales to be when $1150 is spent on advertising? What does the model predict sales to be when $5,000,000 is spent on advertising? way outside But $5,000,000 is way outside the observed values for x. The model should not be used for such predictions.

14. By EXCEL Choose Regression from Data Analysis

15. Check Labels Output Worksheet Location of Y-values X-values

16. b0b0b0b0 b1b1b1b1 Regression Equation Y = 46486.49 + 52.56757x

17. Review minimizing the total sum of the squared errorsb 0, the point estimate for  0, and b 1, the point estimate for  1, are found from calculus by minimizing the total sum of the squared errors between the actual and predicted values of y. The regression equation coefficients can be found by Excel or by hand by: The regression equation should not be used for values of x that are “far away” from the observed x values.