notes Ch 12 - .1 Model Simple Linear Regression The...

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CHAPTER 12 Simple Linear Regression 12.1 Model The regression model used in simple linear regression is: ε β β + + = x y 1 0 0 β and 1 β are the parameters of the model and ε (epsilon) is a random number variable referred to as the error term. The equation that describes how the expected value of y, denoted E(y), is related to x is called the regression equation. x y E 1 0 ) ( β β + = The parameters 0 β and 1 β are typically unknown. Sample statistics (denoted b 0 and b 1 ) are computed as estimates of the population parameters 0 β and 1 β . The estimated simple linear regression equation is a line given by x b b y 1 0 ^ + = Where 0 b is the y-intercept and 1 b is the slope. 12.2 Least Squares Method The least squares method is a procedure for using sample data to find the estimated regression equation. Case: The student population and quarterly sales data for 10 Armand’s Pizza Parlors are provided in the table below. Restaurant Student Population [1000s] Quarterly Sales [$1000s] i x i y i 1 2 58 2 6 105 3 8 88 4 8 118 5 12 117 6 16 137 7 20 157 8 20 169 9 22 149 10 26 202 1
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Based on the data, a scatter diagram of the student population and quarterly sales can be created. Scatter Diagram 0 50 100 150 200 250 0 5 10 15 20 25 30 Student Population [1000s] Quarterly Sales [$1000 The diagram shows a positive relationship between the student population and quarterly sales. The least squares criterion is min 2 ^ ) ( - y y i where i y = observed value of the dependent variable for the ith observation and y = estimated value of the dependent variable for the ith observation Differential calculus can be used to show that values of 0 b and 1 b that minimize the above expression can be found by using the following two equations: x b y b x x y y x x b i i i 1 0 2 1 ) ( ) )( ( - = - - - = Where i x = value of the independent variable for the ith observation i y = value of the dependent variable for the ith observation x = mean value for the independent variable y = mean value for the dependent variable
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