LL8 - Part 8 Linear Regression Number of Cars Sold 30 1....

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BUS1200-8 1 Part 8 Linear Regression 1. Simple Linear Regression Models 2. Multiple Linear Regression Models 3. F Tests 4. Prediction and Cautions 5. Further Inference about Coefficients of Individual Independent Variables Section 8.1 Simple Linear Regression Models Example (Reed Auto) Reed Auto periodically has a special weeklong sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales are shown below. Week 1 2 3 4 5 Number of TV Ads 1 3 2 1 3 Number of Cars Sold 14 24 18 17 27 BUS1200-8 2 By looking at the scatter diagram , we can observe that there exists a strong linear relationship between the number of TV ads (the independent variable , explanatory variable or predictor ) and the number of cars sold (the dependent variable or response variable ). We may draw more than one straight line through the scatter diagram. 0 10 20 30 123 Number of TV Ads Number of Cars Sold Number of TV Ads Number of Cars Sold 0 10 20 30 BUS1200-8 3 Which straight line do you think is the best? Simple linear regression model Assumption 1. For i = 1, 2, …, n , Y i follows N( β 0 + 1 x i , σ 2 ), where 0 and 1 are constants and x i ’s are non -random. n is called the number of observations. 2. Y 1 , Y 2 , …, Y n are independent. Definition For n pairs of data ( x 1 , y 1 ), ( x 2 , Density of Y 1 Density of Y 2 Same shape y x x 2 x 1 0 Regression line y = 0 + 1 x BUS1200-8 4 y 2 ), …, ( x n , y n ), let = = n i i x n x 1 1 , = = n i i y n y 1 1 , b 1 = () = = n i i n i i i x x y y x x 1 2 1 and b 0 = x b y 1 . Then y = b 0 + b 1 x is called the estimated regression equation . Let i y ˆ = b 0 + b 1 x i for i = 1, 2, …, n . Theorem 1. b 0 and b 1 meet the least squares criterion . x 0 x i y ( x i , y i ) i i y y ˆ y = b 0 + b 1 x y i i y ˆ
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BUS1200-8 5 That is, the value of () = n i i i y y 1 2 ˆ will be increased if the value of b 0 or b 1 is made different from that in expression (*). 2. b 0 and b 1 are unbiased estimates of β 0 and 1 respectively. Two formulae for calculation: = n i i i y y x x 1 = = = = n i i n i i n i i i y x n y x 1 1 1 1 , = n i i x x 1 2 = 2 1 1 2 1 = = n i i n i i x n x . Example (Reed Auto) [Solution] i 1 2 3 4 5 Sum x i 1 3 2 1 3 10 y i 14 24 18 17 27 100 x i 2 1 9 4 1 9 24 x i y i 14 72 36 17 81 220 y i 2 196 576 324 289729 2114 BUS1200-8 6 (The last row will be used later.) = n i i i y y x x 1 = = n i i i y x 1 = = n i i n i i y x n 1 1 1 = 220 ) 100 )( 10 ( 5 1 = 20, = n i i x x 1 2 = = n i i x 1 2 2 1 1 = n i i x n = 24 ) 10 ( 5 1 2 = 4, b 1 = 20 / 4 = 5, x b y b 1 0 = = 100 / 5 5 (10 / 5) = 10. [Interpretation] For each increase of one in the number of TV ads, we may say that the number of cars sold is estimated to increase 0 10 20 30 123 Number of TV Ads Number of Cars Sold y = 10 + 5 x BUS1200-8 7 by 5 on the average.
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This note was uploaded on 11/03/2011 for the course ECON 101 taught by Professor Wood during the Spring '07 term at University of California, Berkeley.

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LL8 - Part 8 Linear Regression Number of Cars Sold 30 1....

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