Using R for Introductory Statistics : John Verzani

Using R for introductory statistics 88In the simple linear regression model for describing the relationship between x i andy i , an error term is added to the linear relationship:y i =β 0 +β 1 x i +ε i .(3.2)The value ε i is an error term, and the coefficients β 0 and β 1 are the regressioncoefficients. † The data vector x is called the predictor variable and y the† These are Greek letters: ε is epsilon and β is beta.response variable. The error terms are unknown, as are the regression coefficients. Thegoal of linear regression is to estimate the regression coefficients in a reasonable mannerfrom the data.The term "linear" applies to the way the regression coefficients are used. The modelwould also be considered a linear model. The term “simple” is usedto emphasize that only one predictor variable is used, in contrast with the multipleregression model, which is discussed in Chapter 10.Estimating the intercept β 0 and the slope β 1 gives an estimate for the underlying linearrelationship. We use "hats" to denote the estimates. The estimated regression line is thenwrittenFor each data point x i we have a corresponding value, with being apoint on the estimated regression line.We refer to as the predicted value for y i , and to the estimated regression line as theprediction line. The difference between the true value y i and this predicted value is theresidual, e i :