Module 8 Exercises 1. For each of the plots in Figure 8.1, identify those that suggest (reasonably) that the assump- tions of linear regression are satisfied. For those that don't, identify the single assumption that is most clearly violated by the data for each. nonlinear relationship - linearity assumption violated. (1) 70 60 50 (2) 32 30 28 26 40 24 22 30 20 20 18 . 10 0 1 2 3 4 5 16 0 1 2 3 Assumptions satisfied. 4 5 constant variance assumption violated (3) 35 (4) 34 30 25 20 15 0 1 2 3 4 5 32 30 28 26 24 22 20 18 0 1 2 3 4 Figure 8.1: Data plots. residuals normality assumption violated. 5 distribution of residuals right-skewed.
2. Researchers collected data from 50 pine trees, specifically their height (in metres), age (in years), and whether or not they were producing cones, giving the columns Height, Age, and Coning in their spread sheet. The data were entered into R, and the command Im (Height~ Age, data=Trees) was used to fit a linear model to the (Height, Age) pairs. The (partial) R output is as follows: Coefficients: Estimate Std. Error t value Pr (>|t|) (Intercept) 5.5990 6.082 1.88e-07 Age 3.3949 0.3899 (a) Give the estimated slope and intercept in the linear relationship Height = Bo + ß1Age. Height = 5.5990 + 3.3949 × Age 1 ?, (b) Using the fitted model, estimate how tall a 4-year old tree would be. 5.5990 + 3.949 × 4 = 19.1786 (m)
(c) The following is a quantile plot of the residuals of the fitted model. Does it suggest that the residual variability is normal? Explain why or why not. Normal Q-Q Plot 0 2 - 0 0 1 er Sample Quantiles 00000 0- 00000 0 00000000000 T o 0 -2 -2 -1 0 1 2 Theoretical Quantiles If residuals have a normal distribution, then the normal probability plot should be approximately a straight line. The plot above suggest residual variability follows a normal distribution.