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Unformatted text preview: 1 Chapter 12. Simple Linear Regression and Correlation 12.1 The Simple Linear Regression Model 12.2 Fitting the Regression Line 12.3 Inferences on the Slope Parameter β 1 12.4 Inferences on the Regression Line 12.5 Prediction Intervals for Future Response Values 12.6 The Analysis of Variance Table 12.7 Residual Analysis 12.8 Variable Transformations 12.9 Correlation Analysis 12.10 Supplementary Problems 2 12.1 The Simple Linear Regression Model 12.1.1 Model Definition and Assumptions(1/5) • With the simple linear regression model y i = β + β 1 x i + ε i the observed value of the dependent variable y i is composed of a linear function β + β 1 x i of the explanatory variable x i , together with an error term ε i . The error terms ε 1 , … , ε n are generally taken to be independent observations from a N(0, σ 2 ) distribution, for some error variance σ 2 . This implies that the values y 1 , … ,y n are observations from the independent random variables Y i ~ N ( β + β 1 x i , σ 2 ) as illustrated in Figure 12.1 3 12.1.1 Model Definition and Assumptions(2/5) 4 12.1.1 Model Definition and Assumptions(3/5) • The parameter β is known as the intercept parameter, and the parameter β is known as the intercept parameter , and the parameter β 1 is known as the slope parameter . A third unknown parameter, the error variance σ 2 , can also be estimated from the data set. As illustrated in Figure 12.2, the data values ( x i , y i ) lie closer to the line y = β + β 1 x as the error variance σ 2 decreases. 5 12.1.1 Model Definition and Assumptions(4/5) • The slope parameter β 1 is of particular interest since it indicates how the expected value of the dependent variable depends upon the explanatory variable x , as shown in Figure 12.3 • The data set shown in Figure 12.4 exhibits a quadratic (or at least nonlinear) relationship between the two variables, and it would make no sense to fit a straight line to the data set. 6 12.1.1 Model Definition and Assumptions(5/5) • Simple Linear Regression Model The simple linear regression model y i = β 0 + β 1 x i + ε i fits a straight line through a set of paired data observations (x 1 ,y 1 ), … , (x n , y n ). The error terms ε 1 , … , ε n are taken to be independent observations from a N(0, σ 2 ) distribution. The three unknown parameters, the intercept parameter β 0 , the slope parameter β 1 , and the error variance σ 2 , are estimated from the data set. 7 12.1.2 Examples(1/2) • Example 67 : Car Plant Electricity Usage The manager of a car plant wishes to investigate how the plant ’ s electricity usage depends upon the plant ’ s production. The linear model will allow a month ’ s electrical usage to be estimated as a function of the month ’ s pro duction....
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This note was uploaded on 03/04/2011 for the course MAT 3401 taught by Professor Munther during the Spring '11 term at Cal Poly Pomona.
 Spring '11
 MUNTHER
 Slope, Correlation, Linear Regression

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