10 Math 161B - Spring 2017 1. In a regression analysis of on-the job head injuries of warehouse laborers caused by falling objects, y is a measure of severity of the injury, x1 is an index reflecting both the weight of the object and the distance from which it fell, and x2 and x3 are indicator variables for the nature of head protection worn at the time of the accident, coded as follows: Type of protection None Hard hat Bump cap x2 0 1 0 x3 0 0 1 The regression function used in the study is y = 0 + 1 x1 + 2 x2 + 3 x3 + \u000f For each of the following questions, specify the null hypothesis H0 for the appropriate test in terms of the model parameter(s). (Hint: the null hypothesis should be an equation involving one or more regression slope(s) such as H0 : 1 = 0). (a) For the same object (x1 fixed), does wearing a bump cap reduce the expected severity of the injury as compared to wearing no protection? (b) For the same object (x1 fixed), is the expected severity of injury the same when wearing a hard hat, as when wearing a bump cap? 2. A group of high-technology companies agreed to share employee salary information in an effort to establish salary ranges for technical positions in research and development. The file \"Salary.txt\" contains data obtained for 65 employees: Salary is yearly salary in $k, X1 = Experience is the years of work experience since the last degree obtained, X2 = Supervised is the number of persons currently supervised by the employee, and X30 = Degree is a coded variable for the highest degree obtained (1 = bachelor's degree, 2 = master's degree, 3 = doctoral degree). (a) Is it legitimate to use the coded variable X30 in a multiple linear regression model? Why or why not? (b) Create two indicator variables X3 and X4 for highest degree obtained and enter them into the SPSS file: Degree Bachelor's Master's Doctoral X3 0 1 0 X4 0 0 1 Perform a multiple linear regression of Salary on X1 , . . . , X4 . Write down the estimated model equation. (c) For two employees with the same number of years in experience and the same number of persons supervised, what is the average salary difference between holding a master's and a doctoral degree? (d) Estimate the salary of an employee who supervises three other people, and has two years of work experience after completing his master's degree. 1 Homework 10 Math 161B - Spring 2017 (e) Obtain standardized residuals for the model fitted in (c) and create a normal probability plot for the residuals and a scatter plot of \u000fi against yi . Comment on the appearance of the plots. Are the assumptions we place on the residuals in the model approximately satisfied? (f) Plot the standardized residuals against the predictor variables Experience and Supervised. Which of these two plots causes more concern? Make a suggestion for what could be done to improve the model (you don't have to carry out your suggestions). 2 Salary Experience Supervised 58.8 4.49 0 34.8 2.92 0 163.7 29.54 42 70 9.92 0 55.5 0.14 0 85 15.96 4 34 2.27 0 29.7 1.2 0 56.1 5.33 3 70.6 15.74 0 74.2 22.46 2 34.1 3.16 0 31.6 2.62 0 65.5 15.06 5 57.2 2.92 0 60.3 2.26 0 41.8 9.76 1 76.5 12.71 4 122.1 21.76 10 85.9 15.63 8 55.9 1.17 0 44.3 2.33 0 79.9 17.1 18 58.5 7.45 0 57.3 4.55 0 61 14.39 8 52.2 5.78 3 45.7 2.08 0 44.8 1.44 0 39.1 1 0 68.1 10.53 5 48.2 19.23 0 51 5.18 0 40.7 4.43 0 51.4 3.04 0 40.9 1.02 0 57.7 10.14 5 95.5 26.53 8 43.9 6.49 0 66.6 13.97 7 30 4.18 0 64.9 12.88 6 151.2 16.01 28 72.4 11.13 6 41.8 0.71 0 Degree 3 1 3 3 3 2 1 1 2 3 1 1 1 1 3 3 1 3 3 3 3 2 3 2 3 2 2 2 2 2 2 2 2 1 2 2 1 3 1 2 1 3 2 2 2 57.8 72.7 36.1 39.8 29 40.4 40.7 41.7 97.2 85.3 42.6 39.1 46.6 53.9 87.4 81.7 42.5 40 60.5 104.8 1.55 3.92 4.37 0.79 0.65 0.69 1.09 1.58 10.89 21.08 7 4.09 8.86 11.05 2.37 6.37 8 0.44 2.1 19.81 0 0 0 0 0 0 0 0 8 0 0 0 0 6 13 0 0 0 0 24 3 3 1 2 1 2 2 2 3 2 2 1 2 2 3 3 1 2 3 3