# Ch17 - Chapter 17 Simple Linear Simple Regression 1 17.1...

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1 Simple Linear Simple Linear Regression Regression Chapter 17

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2 17.1 Introduction In Chapters 17 to 19 we examine the relationship between interval variables via a mathematical equation. The motivation for using the technique: Forecast the value of a dependent variable (y) from the value of independent variables (x 1 , x 2 ,…x k .). Analyze the specific relationships between
3 House size House Cost Most lots sell for \$25,000 Building a house costs about \$75 per square foot. House cost = 25000 + 75(Size) 17.2 The Model The model has a deterministic and a probabilistic components

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4 House cost = 25000 + 75(Size) House size House Cost Most lots sell for \$25,000 + ε However, house cost vary even among same size houses! 17.2 The Model Since cost behave unpredictably, we add a random component.
5 17.2 The Model The first order linear model y = dependent variable x = independent variable β 0 = y-intercept β 1 = slope of the line ε = error variable ε + β + β = x y 1 0 x y β 0 Run Rise β 1 = Rise/Run β 0 and β 1 are unknown population parameters, therefore are estimated from the data.

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6 17.3 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.      Question: What should be considered a good line? x y
7 The Least Squares (Regression) Line A good line is one that minimizes the sum of squared differences between the points and the line.

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8 The Least Squares (Regression) Line 3 3 4 1 1 4 (1,2) 2 2 (2,4) (3,1.5) Sum of squared differences = (2 - 1) 2 + (4 - 2) 2 + (1.5 - 3) 2 + (4,3.2) (3.2 - 4) 2 = 6.89 Sum of squared differences = (2 -2.5) 2 + (4 - 2.5) 2 + (1.5 - 2.5) 2 + (3.2 - 2.5) 2 = 3.99 2.5 Let us compare two lines The second line is horizontal The smaller the sum of squared differences the better the fit of the line to the data.
9 The Estimated Coefficients To calculate the estimates of the line coefficients, that minimize the differences between the data points and the line, use the formulas: x b y b s ) Y , X cov( b 1 0 2 x 1 - = = The regression equation that estimates the equation of the first order linear mod is: x b b y ˆ 1 0 + =

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10 Example 17.2 ( Xm17-02 ) 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, CarOdometer Price 1 37388 14636 2 44758 14122 3 45833 14016 4 30862 15590 5 31705 15568 6 34010 14718 . . . . . . . . . Independent variable x Dependent variable y The Simple Linear Regression Line
11 The Simple Linear Regression Line Solution Solving by hand: Calculate a number of statistics ; 823 . 822 , 14 y ; 45 . 009 , 36 x = = 511 , 712 , 2 1 n ) y y )( x x ( ) Y , X cov( 690 , 528 , 43 1 n ) x x ( s i i 2 i 2 x - = - - - = = - - = where n = 100. 067 , 17 ) 45 . 009 , 36 )( 06232 . ( 82 . 822 , 14 x b y b 06232 . 690 , 528 , 43 511 , 712 , 1 s ) Y , X cov( b 1 0 2 x 1 = - - = - = - = - = = x 0623 . 067 , 17 x b b y ˆ 1 0 - = + =

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12 Solution – continued Using the computer ( Xm17 -02) Tools > Data Analysis > Regression > [Shade the y range and the x range] > OK The Simple Linear Regression Line
13 SUMMARY OUTPUT Regression Statistics Multiple R 0.8063 R Square 0.6501 Adjusted R 0.6466 Standard E 303.1 Observatio 100 ANOVA df SS MS F Significance F Regression 1 16734111 16734111 182.11 0.0000

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Ch17 - Chapter 17 Simple Linear Simple Regression 1 17.1...

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