CHAPTER 12 SimpleLinearRegression

CHAPTER 12 SimpleLinearRegression - Goldsman ISyE 6739...

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Goldsman — ISyE 6739 Linear Regression REGRESSION 12.1 Simple Linear Regression Model 12.2 Fitting the Regression Line 12.3 Inferences on the Slope Parameter 1
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Goldsman — ISyE 6739 12.1 Simple Linear Regression Model Suppose we have a data set with the following paired observations: ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) Example: x i = height of person i y i = weight of person i Can we make a model expressing y i as a function of x i ? 2
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Goldsman — ISyE 6739 12.1 Simple Linear Regression Model Estimate y i for fixed x i . Let’s model this with the simple linear regression equation, y i = β 0 + β 1 x i + ε i , where β 0 and β 1 are unknown constants and the error terms are usually assumed to be ε 1 , . . . , ε n iid N (0 , σ 2 ) y i N ( β 0 + β 1 x i , σ 2 ) . 3
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Goldsman — ISyE 6739 12.1 Simple Linear Regression Model y = β 0 + β 1 x with “high” σ 2 y = β 0 + β 1 x with “low” σ 2 4
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Goldsman — ISyE 6739 12.1 Simple Linear Regression Model Warning! Look at data before you fit a line to it: doesn’t look very linear! 5
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Goldsman — ISyE 6739 12.1 Simple Linear Regression Model x i y i Production Electric Usage ($ million) (million kWh) Jan 4.5 2.5 Feb 3.6 2.3 Mar 4.3 2.5 Apr 5.1 2.8 May 5.6 3.0 Jun 5.0 3.1 Jul 5.3 3.2 Aug 5.8 3.5 Sep 4.7 3.0 Oct 5.6 3.3 Nov 4.9 2.7 Dec 4.2 2.5 6
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Goldsman — ISyE 6739 12.1 Simple Linear Regression Model 3.5 4.0 4.5 5.0 5.5 6.0 2.2 2.6 3.0 3.4 x i y i Great... but how do you fit the line? 7
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Goldsman — ISyE 6739 12.2 Fitting the Regression Line Fit the regression line y = β 0 + β 1 x to the data ( x 1 , y 1 ) , . . . , ( x n , y n ) by finding the “best” match between the line and the data. The “best”choice of β 0 , β 1 will be chosen to minimize Q = n summationdisplay i =1 ( y i - ( β 0 + β 1 x i )) 2 = n summationdisplay i =1 ε 2 i .
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