Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 19 Linear Regression Moo K. Chung [email protected] Nov 30, 2004 Concepts 1. Let x be the speed of a car and y be the distance the car traveled in an hour hour. Then we have model y = β 0 + β 1 x. Suppose we have n paired measurements ( x i , y i ) , i = 1 , · · · , n . Since all measurement are supposed to be noisy, we introduce a noise term ² in the above equation. Our modified stochastic model is y = β 0 + β 1 x + ², where ² N (0 , σ 2 ) . Since ² is a random variable, we use Y instead of y for convenience: Y = β 0 + β 1 x + ². Note that E Y = β 0 + β 1 x and V Y = σ 2 . 2. Equivalently we can write the above linear model for each paired measurement ( x i , y j ) : Y j = β 0 + β 1 x j + ² j , where y j is the observed value of random variable Y j and ² j ² . Note that E Y j = β 0 + β 1 x j . Let ˆ β 0 , ˆ β 1 be estimators of β 0 , β 1 . Then the predicted values or fitted values are given by ˆ y j = ˆ β 0 + ˆ β 1 x j . The differences between the observations y j and the pre- dicted values ˆ y j are called the residuals (errors), i.e. r j = y j - ˆ y j = y j - ˆ β 0 - ˆ β 1 x j . 3. Least squares estimation . The least squares estimation is a method of estimating parameters β 0 and β 1 by min- imizing the sum of the squared errors (SSE):
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