1 Simple Linear Regression and Correlation Least Squares Method The Model Estimating the Coefficients EXAMPLE 1: USED CAR SALES

2 The Least Squares Method –We are seeking a line that best fit the data –We define “best fit line” as a line for which the sum of squared differences between it and the data points is minimized. The actual y value of point i The y value of point i calculated from the equation of the line

3 Errors Different lines generate different errors, thus different sum of squares of errors. X Y

4 The coefficients b 0 and b 1 of the line that minimizes the sum of squares of errors are calculated from the data.

5 The Model The first order linear model y = dependent variable x = independent variable  0 = y-intercept  1 = slope of the line = error variable x y 00 Run Rise   = Rise/Run  0 and  1 are unknown, therefore, are estimated from the data.

6 Estimating the Coefficients The estimates are determined by –drawing a sample from the population of interest, –calculating sample statistics. –producing a straight line that cuts into the data.           The question is: Which straight line fits best? x y

7 3 3 The best line is the one that minimizes the sum of squared vertical differences between the points and the line.     (1,2) 2 2 (2,4) (3,1.5) Sum of squared differences =(2 - 1) 2 +(4 - 2) 2 +( ) 2 + (4,3.2) ( ) 2 = The smaller the sum of squared differences the better the fit of the line to the data. X Y

8 To calculate the estimates of the coefficients that minimize the differences between the data points and the line, use the formulas: The regression equation that estimates the equation of the first order linear model is:

9 Example 1 Relationship between odometer reading and a used car’s selling price. –A car dealer wants to find the relationship between the odometer reading and the selling price of used cars. –A random sample of 100 cars is selected, and the data recorded. –Find the regression line. Independent variable x Dependent variable y

10 Solution –Solving by hand To calculate b 0 and b 1 we need to calculate several statistics first; where n = 100.