lecture20

# lecture20 - Statistical inference for straight-line model...

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Statistical inference for straight-line model Recall the model y = β 0 + β 1 x + ε The least squares estimators for β 1 and β 0 are given by ˆ β 1 = SS xy SS xx , ˆ β 0 = ¯ y ˆ β 1 ¯ x By fitting this model, we know the estimates are too much focused on the sample data. Can we say something about these estimates on different samples? In other words, how reliable are they?

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Assumptions about the random component ε ε is assumed to have a probability distribution with mean 0 The variance of the probability distribution of ε is constant for all settings of the independent x . By notation, we assume Var[ ε ] σ 2 . ε has a normal distribution. The values of ε associated with any two observed values of y are independent.
Estimating σ 2 Recall SSE= ( y i ˆ y i ) 2 , expanding it out and by some algebra SSE = SS yy ˆ β 1 SS xy in which SS yy = summationdisplay y 2 i ( y i ) 2 n One estimator for σ 2 is given by s 2 = SSE n 2 where n is the sample size.

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