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

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Stat 312: Lecture 17 Simple Linear Model Moo K. Chung [email protected] March 27, 2003 Concepts 1. Deterministic model: y = β 0 + β 1 x. For example, x is the speed of a car and y is the distance the car traveled in 1 hour. This is an unrealistic model because the car can not maintain the absolute constant speed. So we introduce a noise term in the above equa- tion. 2. Stochastic model: let be a random noise with = 0 and Var = σ 2 . Instead of the deterministic model we will have y = β 0 + β 1 x + . Since is a random variable, we use Y instead of y : Y = β 0 + β 1 x + . Note that Y = β 0 + β 1 x and Var Y = σ 2 . 3. Given paired data ( x 1 , y 1 ) , · · · , ( x n , y n ) , we have a lin- ear stochastic relationship Y j = β 0 + β 1 x j + j , where y j is the observed value of a random variable Y j and j . Note that 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 ˆ y j = ˆ β 0 + ˆ β 1 x j are estimators of Y j = β 0 + β 1 x j . The differences between the observations y j and the predicted 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 . 4. Least squares estimation . We estimate β 0 and β 1 by minimizing the sum of the squared errors (SSE): SSE = n X j
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