Lecture21 - lecture 21 Linear Regression II lecture 21...

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lecture 21, Linear Regression II 1 / 24 Standard Error Testing the Signiﬁcance of the Slope Reading the Computer Output lecture 21, Linear Regression II Outline 1 Standard Error 2 Testing the Signiﬁcance of the Slope 3 Reading the Computer Output

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lecture 21, Linear Regression II 2 / 24 Standard Error Testing the Signiﬁcance of the Slope Reading the Computer Output lecture 21, Linear Regression II Review Simple linear regression model: y = β 0 + β 1 x + ε, ε is N ( 0 , σ 2 ) . ( y = μ y | x + ε, ε is N ( 0 , σ 2 ) , μ y | x = β 0 + β 1 x . b 0 , b 1 : point estimates of β 0 , β 1 Estimation/prediction equation: ˆ y = b 0 + b 1 x (1)
lecture 21, Linear Regression II 3 / 24 Standard Error Testing the Signiﬁcance of the Slope Reading the Computer Output lecture 21, Linear Regression II Review, ctd The least squares line: the line that minimizes the sum of squared error (SSE) (i.e., sum of squared residuals) SSE = n X i = 1 ( y i - ˆ y i ) 2 , where ˆ y i = b 0 + b 1 x i Least squares point estimates of β 0 and β 1 : b 1 = SXY SXX := n i = 1 ( x i - ¯ x )( y i - ¯ y ) n i = 1 ( x i - ¯ x ) 2 , b 0 = ¯ y - b 1 ¯ x .

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lecture 21, Linear Regression II 4 / 24 Standard Error Testing the Signiﬁcance of the Slope Reading the Computer Output lecture 21, Linear Regression II Estimating a Mean Fuel Consumption The least squares estimates of β 0 , β 1 : b 0 = 15 . 84 and b 1 = - 0 . 1279 . The least squares line: ˆ y = 15 . 84 - 0 . 1279 x . What’s the mean fuel consumption when the temperature x = 40? We want to estimate μ y | x = 40 = β 0 + β 1 · 40 , A point estimate: ˆ y = = 10 . 72 .
lecture 21, Linear Regression II 5 / 24 Standard Error Testing the Signiﬁcance of the Slope Reading the Computer Output lecture 21, Linear Regression II Predicting an Individual Fuel Consumption What’s the fuel consumption that you would predict when the temperature x = 40? The fuel consumption when the temperature x = 40 is y = β 0 + β 1 · 40 + ε We have seen that ˆ y = b 0 + b 1 · 40 is a point estimate of β 0 + β 1 · 40 How about ε ? By assumption, ε is a normal random variable with mean zero, so it is symmetric around 0, and it is reasonable to predict the error term ε to be Point prediction: ˆ y = 15 . 84 - 0 . 1279 · 40 + 0 = 10 . 72

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lecture 21, Linear Regression II 6 / 24 Standard Error Testing the Signiﬁcance of the Slope Reading the Computer Output lecture 21, Linear Regression II Point Estimation and Point Prediction
lecture 21, Linear Regression II 7 / 24 Standard Error Testing the Signiﬁcance of the Slope Reading the Computer Output lecture 21, Linear Regression II Point Estimation and Point Prediction, ctd Experimental region is the range of the observed values of x Suppose that the least squares line is ˆ y = b 0 + b 1 x .

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Lecture21 - lecture 21 Linear Regression II lecture 21...

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