sta108_handout1

sta108_handout1 - Handout 1 Simple linear regression model...

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Unformatted text preview: Handout 1 Simple linear regression model: The data consist of n pairs of observations ( X 1 ,Y 1 ) ,..., ( X n ,Y n ). The model is Y i = β + β 1 X i + ε i ,i = 1 ,...,n, where ε 1 ,...,ε n are independent, E ( ε i ) = 0 and V ar ( ε i ) = σ 2 . From this it follows that E ( Y i ) = β + β 1 X i , V ar ( Y i ) = σ 2 and that Y 1 ,...,Y n are independent. This model can be interpreted in the following way. In the population, the mean and variance of the Y ’s in the vertical strip at X are β + β 1 X and σ 2 , respectively. In other words, the means of the vertical strips lie on a straight line with intercept β and slope β 1 . However, according to this model, the variances of the vertical strips are the same, namely σ 2 . Note that these are assumptions of the model, and appropriateness of these assumptions can be checked by statistical methods which are discussed in chapter 3. Estimation of β and β 1 (least squares method) For this method we find the straight line (equivalent to finding its intercept b and its slope β 1 ) which fits the data the closest. Let Q = X 1 ≤ i ≤ n ( Y i- b- b 1 X i ) 2 ....
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sta108_handout1 - Handout 1 Simple linear regression model...

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