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Chapter 12 - MAT2378 Rafal Kulik Version 2009/Nov/23 Rafal...

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MAT2378 Rafal Kulik Version 2009/Nov/23 Rafal Kulik
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MAT2378 Probability and Statistics for the Natural Sciences Chapter 12 Comments These notes cover material from Chapter 12. I’m planning to spend two lectures on this material. Rafal Kulik 1
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MAT2378 Probability and Statistics for the Natural Sciences Chapter 12 Rafal Kulik 2
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MAT2378 Probability and Statistics for the Natural Sciences Chapter 12 Regression Analysis We want to describe the relationship between the predictor , X - Body Weight and the response variable , Y - number of mature eggs. We will use regression analysis . We will assume that our model is given by Y = β 0 + β 1 X + , where is a random error and β 0 , β 1 are regression coefficients . It is assumed that E( ) = 0 , Var( ) = σ 2 . This implies that μ Y | X = β 0 + β 1 X σ Y | X = σ, where μ Y | X is the population mean of Y for a given X , σ Y | X is the population standard deviation of Y for a given X . Rafal Kulik 3
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MAT2378 Probability and Statistics for the Natural Sciences Chapter 12 The parameters β 0 , β 1 are unknown and have to be estimated by b 0 , b 1 . Consequently, we will find the (fitted) regression line or the line of the best fit : ˆ Y = b 0 + b 1 X. Rafal Kulik 4
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MAT2378 Probability and Statistics for the Natural Sciences Chapter 12 Least squares estimation Suppose now that we have observations ( x i , y i ) from our model, so y i = β 0 + β 1 x i + i , i = 1 , . . . , n. Our aim is to find b 0 , b 1 , estimators of the unknown parameters β 0 , β 1 : b 1 = S xy S xx , b 0 = ¯ y - b 1 ¯ x, where S xy = n i =1 ( x i - ¯ x )( y i - ¯ y ) S xx = n i =1 ( x i - ¯ x ) 2 S yy = n i =1 ( y i - ¯ y ) 2 Rafal Kulik 5
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MAT2378 Probability and Statistics for the Natural Sciences Chapter 12 Residuals This line is obtained using the method of least squares . Having the observations y i , i = 1 , . . . , n , their deviations from the line ˆ y = b 0 + b 1 x are y i - ˆ y i = ( y i - b 0 - b 1 x i ) , i = 1 , . . . , n. The total deviation of the observed values y i from the estimated values ˆ y i is measured using Sum of Squares of Residuals : SS(resid) = n i =1 ( y i - ˆ y i ) 2 . From this, we can estimate σ : s Y | X = ˆ σ = SS(resid) n - 2 . Rafal Kulik 6
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MAT2378 Probability and Statistics for the Natural Sciences Chapter 12 Example For eggs data we find The regression equation is eggs = - 72.1 + 31.8 body-wt S = 22.6010 R-Sq = 47.3% R-Sq(adj) = 45.9% Analysis of Variance Source DF SS MS F P Regression 1 16980 16980 33.24 0.000 Residual Error 37 18900 511 Total 38 35879 This means: b 0 = - 72 . 1 , b 1 = 31 . 8
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