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**Unformatted text preview: **MAT2378 Rafal Kulik Version 2009/Nov/23 Rafal Kulik 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 MAT2378 Probability and Statistics for the Natural Sciences Chapter 12 Rafal Kulik 2 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 + β 1 X + , where is a random error and β , β 1 are regression coefficients . It is assumed that E( ) = 0 , Var( ) = σ 2 . This implies that μ Y | X = β + 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 MAT2378 Probability and Statistics for the Natural Sciences Chapter 12 The parameters β , β 1 are unknown and have to be estimated by b , b 1 . Consequently, we will find the (fitted) regression line or the line of the best fit : ˆ Y = b + b 1 X. Rafal Kulik 4 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 + β 1 x i + i , i = 1 , . . . , n. Our aim is to find b , b 1 , estimators of the unknown parameters β , β 1 : b 1 = S xy S xx , b = ¯ y- b 1 ¯ x, where S xy = n X i =1 ( x i- ¯ x )( y i- ¯ y ) S xx = n X i =1 ( x i- ¯ x ) 2 S yy = n X i =1 ( y i- ¯ y ) 2 Rafal Kulik 5 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 + b 1 x are y i- ˆ y i = ( y i- b- 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 X i =1 ( y i- ˆ y i ) 2 ....

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