Multiple Linear Regression

2.8 How to do it with RTo get confidence intervals for the parameters we need only use confint:> confint(trees.lm)2.5 % 97.5 %(Intercept) -75.68226247 -40.2930554Girth 4.16683899 5.2494820Height 0.07264863 0.6058538For example, using the calculations above we say that for the regression model Volume~Girth + Height we are 95% confident that the parameter β 1 lies somewhere in the interval[4.2, 5.2].2.9 Confidence and Prediction IntervalsWe use confidence and prediction intervals to gauge the accuracy of our parameter estimates.We know Ŷ(x 0 ) = x T 0b, and we also know(b ∼ mvnorm mean = β, sigma = σ ( 2 X T X ) ) −1, (27)so we get(Ŷ(x 0 ) ∼ mvnorm mean = x T 0 β, sigma = ( σ2 x T 0 X T X ) )−1x0 . (28)and confidence intervals for the mean value of a future observation at the location x 0 = [ x 10 x 20 . . . x p0] Tare given byŶ(x 0 ) ± t α/2 (df = n − p − 1) SPrediction intervals for a new observation at x 0 are given byŶ(x 0 ) ± t α/2 (df = n − p − 1) S√x T (0 XT X ) −1 x 0 . (29)√1 + x T (0 XT X ) −1 x 0 . (30)Note that the prediction intervals are wider than the confidence intervals.2.10 How to do it with R> new