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chapter13

# chapter13 - Introduction to Probability and Statistics and...

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Introduction to Probability Introduction to Probability and Statistics and Statistics Thirteenth Edition Thirteenth Edition Chapter 13 Multiple Regression Analysis

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Introduction Introduction We extend the concept of simple linear regression as we investigate a response y which is affected by several independent variables, x 1 , x 2 , x 3 ,…, x k . Our objective is to use the information provided by the x i to predict the value of y.
Example Example Let y be a student’s college achievement, measured by his/her GPA. This might be a function of several variables: x 1 = rank in high school class x 2 = high school’s overall rating x 3 = high school GPA 4

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Example Example Let y be the monthly sales revenue for a company. This might be a function of several variables: x 1 = advertising expenditure x 2 = time of year x 3 = state of economy 4
Some Questions Some Questions How well does the model fit? How strong is the relationship between y and the predictor variables? Have any assumptions been violated? How good are the estimates and predictions? We collect information using n observations on the response y and the independent variables, x , x , x , … x .

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The General The General Linear Model Linear Model
The Random Error The Random Error The deterministic part of the model, E(y) = E(y) = β β + β + β x x + + β β x x +…+ +…+ β β x x , , describes average value of y for any fixed

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Example Example Consider the model E( y ) = β 0 + β 1 x 1 + β 2 x 2 This is a first order model first order model (independent variables appear only to the first power). β 0 = y y -intercept -intercept = value of E( y ) when x 1 = x 2 =0. β 1 and β 2 are the partial regression partial regression coefficients coefficients —the change in y for a one- unit change in x i when the other when the other independent variables are held constant independent variables are held constant . Traces a plane plane in three dimensional space.
The Method of The Method of Least Squares Least Squares The best-fitting prediction equation is calculated using a set of n measurements ( y , x 1 , x 2 ,… x k ) as We choose our estimates b 0 , b 1 ,…, b k to estimate β 0 , β 1 ,…, β k to minimize 2 1 1 0 2 ) ... ( ) ˆ ( k k x b x b b y y y - - - - = - = SSE k k x b x b b y + + + = ... ˆ 1 1 0

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Example Example A computer database in a small community contains the listed selling price y (in thousands of dollars), 4 Property y Fit a first order model to the data using the method of least squares.
Example Example The first order model is E( y ) = β + β x + β x + β x + β x fit using Minitab with the values of y and the four independent variables entered into five Regression Analysis: ListPrice versus SqFeet, NumFlrs, Bdrms, Baths The regression equation is ListPrice = 18.8 + 6.27 SqFeet - 16.2 NumFlrs - 2.67 Bdrms + 30.3 Baths Predictor Coef SE Coef T P Constant 18.763 9.207 2.04 0.069 SqFeet 6.2698 0.7252 8.65 0.000 NumFlrs -16.203 6.212 -2.61 0.026 Bdrms -2.673 4.494 -0.59 0.565 Baths 30.271 6.849 4.42 0.001 Partial regression coefficients Regression equation

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The total variation in the experiment is measured by the total sum of squares
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