Chapter 12. Simple Linear Regression and Correlation
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 Rarameter 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
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12.1 The Simple Linear Regression Model
12.1.1 Model Definition and Assumptions(1/5)
? With the simple linear regression model yi=0+1xi+i the observed value of the dependent variable yi is composed of a linear function 0+1xi of the explanatory variable xi, together with an error term i. The error terms 1,...,n are generally taken to be independent observations from a N(0,2) distribution, for some error variance 2. This implies that the values y1,...,yn are observations from the independent random variables Yi ~ N (0+1xi, 2) as illustrated in Figure 12.1
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12.1.1 Model Definition and Assumptions(2/5)
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12.1.1 Model Definition and Assumptions(3/5)
? The parameter 0 is known as the intercept parameter, and the parameter 0 is known as the intercept parameter, and the parameter 1 is known as the slope parameter. A third unknown parameter, the error variance 2, can also be estimated from the data set. As illustrated in Figure 12.2, the data values (xi , yi ) lie closer to the line y = 0+1x as the error variance 2 decreases.
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12.1.1 Model Definition and Assumptions(4/5)
? The slope parameter 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.
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