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# notes3b - Notes on regression 1 Simple Linear Regression...

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: 1. Simple Linear Regression Model: Y i = β 0 + β 1 X i + ± i , i = 1 , ..., n where: Y i is the value of the response variable in the i th trial β 0 and β 1 are parameters X i is a known constant, namely, the value of the independent variable in the i th trial ± i is an error term with mean 0 and variance σ 2 ; ± i and ± j are uncorrelated for i 6 = j . (Usually, we also assume that ± i is normally distributed.) The least squares estimates of β 1 and β 0 are respectively given by ˆ β 1 = n i =1 ( X i - ¯ X )( Y i - ¯ Y ) n i =1 ( X i - ¯ X ) 2 and ˆ β 0 = ¯ Y - ˆ β 1 ¯ X. where ¯ X = n i =1 X i /n and ¯ Y = n i =1 Y i /n . The ﬁtted (or estimated) regression line is given by ˆ Y = ˆ β 0 + ˆ β 1 X For the observations in the sample, we will call ˆ Y i : ˆ Y i = ˆ β 0 + ˆ β 1 X i the ﬁtted value for the i th observation ( i = 1 , ..., n ). A natural measure of the eﬀect of X in reducing the variation in Y , i.e., the uncertainty in predicting

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notes3b - Notes on regression 1 Simple Linear Regression...

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