Using R for Introductory Statistics : John Verzani

Linear regression 275Figure 10.5 Four diagnostic plots forthe maximum-heart-rate dataproduced by the extractor functionplot ()10.2.2 Statistical inferencesIf the linear model seems appropriate for the data, statistical inference is possible. What isneeded is an understanding of the sampling distribution of the estimators.To investigate these sampling distributions, we performed simulations of the modelY i =x i +ε i , using x=rep(1:10,10) and y=rnorm(100,x,5). Figure 10.6 shows the resultingregression lines for the different simulations. For reference, a single result of thesimulation is plotted using a scatterplot. There is wide variation among the regressionlines. In addition, histograms of the simulated values of and are shown.We see from the figure that the estimators are random but not arbitrary. Both andare normally distributed, with respective means β 0 and β 1 . Furthermore,has a χ 2 -distribution with n−2 degrees of freedom.We will use the fact that the following statistics have a t-distribution with n−2 degreesof freedom: