3 - STAT 462, Fall 2015 Please bring a hard copy of your homework to class and submit it at the beginning of the class or submit it via ANGEL before class. You could either type or write down your answers. Attach your code used to obtain your results. This is should be as neat as possible. Make sure your name is on it. Deadline: 09/25/2015. Exercise 1 (25 points) Load into R Iris data using the following command data(iris). the data consists of meaurements taken from 50 owers for each of 3 plant species, resulting in 150 observations. Let be y =Petal.Width and x =Petal.Length. 1. Compute in R, the estimates b0 , b1 and s2 . (You can use the function lm in R if you want). 2. Are 0 and 1 signicantly dierent from 0? What can you conclude? Make sure your conclusions reference to the data application. 3. Compute the ANOVA table including the F-test and the corresponding p-value. Decide whether or not there is a linear association between Petal.Width and Petal.Length. Exercise 2 (25 points) Load into R Iris data using the following command data(iris). The data set contains 3 classes of 50 instances each, where each class refers to a type of iris plant. Let be y =Petal.Width and x =Petal.Length. Given the output of exercise 1: 1. Create a residual plot against the tted values. Does the linearity assumption hold? What about equivariance? 2. Construct a histogram of the residuals, do they look normal? 3. Create a normal probability plot of the residuals. What are your conclusions? Exercise 3 (25 points) Using the R package datasets - library(datasets) - load in R the dataset called faithful. N = 272 eruptions of the Old Faithful geyser in the United States are used to study the straight-line relationship between y = time to next eruption (min.) and x = duration of the present eruption (min.). Run in R the following commands: library(datasets) attach(faithful) t=lm(waiting eruptions) t summary(t) par(mfrow=c(1,2)) plot.lm(t,1:2) 1. State the assumptions of a SLR model. 2. Describe what each plot is and what it is for. 3. Based on the plots, do the modeling assumptions appear to be met? 1 Exercise 4 (25 points) The following data were obtained in a study of the relation between diastolic blood pressure (Y) and age (X) for boys 5 to 13 years old. i xi yi Table 1: Blood pressure 1 2 3 4 5 6 7 5 8 11 7 13 12 12 63 67 74 64 75 69 90 8 6 60 1. After tting the model, plot the residuals against the tted values 2. Omit case 7 and ret the model. Plot the residuals versus the tted values and compare to what you got in (1). 3. Using the tted model from (2), obtain a 99% prediction interval for a new Y observation at X = 12. Does observation y7 fall outside this prediction interval? What is the signicance of this? 2