3. (28 points) Consider a data set of 173 random female horseshoes crabs where the weight (gram), the carapace width (cm) and the spine condition of each crab were recorded. The spine condition is divided into three groups: both good (1); one worn or broken (2) and both worn or broken (3). We are interested in explaining the weight of the crabs based on their carapace width and spine condition. (a) A histogram of the variable weight and a scatter plot of weight vs width are given in Figure 1. Comment on the suitability of tting a linear regression model for the variable weight with regressor width. Histogram of weight Density 0.0 0.1 0.2 0.3 0.4 0.5 0.6 weight 2 3 22 24 26 28 30 32 34 weight width Figure 1: Plots before fitting a model PART I A simple linear regression model is fitted (called Model 1), where the variable width is the regressor. The coefficients of Model 1 is given in Figure 2. The residual plots and some further outputs of Model 1 are given in Figure 3 and Figure 4. (b) Conduct a detailed check of the adequacy of Model 1 (check model assumptions using residual plots, and check for outliers and influential points). (c) Test the significance of Model 1 at significance level 0.01. Coefficients : Estimate Std. Error (Intercept) -3944 . 019 255. 027 width 242 . 642 9 . 666 Figure 2: Coefficients of Model 1QQ Plot of SR of Model 1 Weight~Width o V Standardized Residuals Z scores oo O 2 1500 2000 2500 3000 3500 4000 Standardized Residuals Fitted Weight Figure 3: Residual plots of Model 1 > SR = rstandard (modell) # Model 1 > C = cooks . distance (model1) > which (C>1) named integer (0) > which (SR >3) 141 141 > which (SR model bw 11918958 1941.3 - spine 2 286383 12205341 1941.4 - width 1 39241243 51160202 2191.3 Step: AIC=1935.38 weight ~ width + spine Df Sum of Sq RSS AIC 11925667 1935. 4 - spine 2 298873 12224540 1935.7 - width 1 41788210 53713877 2193.7 Figure 7: Backward Variable Selection