1 IHI Insurance have conducted a multiple linear regression analysis to predict the value of the number of annual claims incurred by a motor insurance policyholder (y)based on the age of the policyholder in years (x1) and the value of the car insured in thousands of dollars (x2). The analysis was based on a random sample of 500 policyholders. The ages in the sample ranged from 16 to 70 and the values of the cars in the sample ranged from $1,100 to $52,000. The multiple linear regression equation corresponding to IHI's analysis is: y^i = 13 + (-0.5)x1i + (0.1)x2i. Select whether the following statements regarding the multiple linear regression equation are true or false: True a) Holding x1 constant, a one thousand dollar increase in x2 will result in a decrease of 0.5 in the predicted value of y. b) A 22 year old policyholder with a car worth $20,000 is predicted to make 4 claims. c) Holding x2 constant, a one year increase in x1 will result in an increase of 0.1 in the predicted value of Y. False PROBLEM 2 A regression model is constructed with the goal of predicting the number of motor vehicle accidents in a city per year based upon the population of the city, the number of recorded traffic offenses per year, the number of vehicles per capita in the city and the average annual temperature in the town. A random sample of 50 cities were studied for this purpose. Here is an analysis output on the regression model: ANOVA DF SS MS F Probability Regression 4 161.359 40.33975 10.72878373... < 0.001 Residual 45 169.198 3.75995556... Total 49 330.557 Regression analysis R2 0.48814274... s 1.93906048... Regression coefficients Estimate Standard Error t Probability Intercept 19.70 2.852 6.90743338... < 0.001 Population of city 2.496 0.1337 18.66866118... < 0.001 No. of vehicles per capita 1.473 0.2480 5.93951613... < 0.001 No. of traffic offenses 0.293 0.4064 0.72096457... 0.47466038... Average annual temp. 0.403 0.4530 0.88962472... 0.37839879... a)At a level of significance of 0.05, the result of the F test for this model is that the null hypothesis rejected. b)Suppose you are going to construct a new model by removing the most insignificant variable. You would first remove: population of city no. of vehicles per capita no. of traffic offenses average annual temp. [14 points]- 3 of 8 ID: MST.MR.CM.01.0020 A multiple regression model is to be constructed to predict the final exam score of a university student doing a particular course based upon their mid-term exam score, the average number of hours spent studying per week and the average number of hours spent watching television per week. Data has been collected on 30 randomly selected individuals: show data a)Find the multiple regression equation using all three explanatory variables. Assume that x 1 is mid-term score, x2 is hours studying per week and x3 is hours watching television per week. Give your answers to 3 decimal places. y^ = + mid-term score + hours studying + hours watching television b)At a level of significance of 0.05, the result of the F test for this model is that the null hypothesis rejected. c)The explanatory variable that is most correlated with final score is: mid-term score hours studying per week hours watching television per week d)The explanatory variable that is least correlated with final score is: mid-term score hours studying per week hours watching television per week e)The value of R2 for this model, to 2 decimal places, is equal to f)The value of s for this model, to 3 decimal places, is equal to g)Construct a new multiple regression model by removing the variable average hours spent watching television per week. Give your answers to 3 decimal places. The new regression model equation is: y^ = + mid-term score + hours studying h)In the new model compared to the previous one, the value of R 2 (to 2 decimal places) is: increased decreased unchanged i)In the new model compared to the previous one, the value of s (to 3 decimal places) is: increased decreased unchanged [1 point]- 4 of 8 ID: MST.MR.MRE.03.0010 Geoff is about to open a restaurant. He has found two possible locations for the restaurant; one closer to the Central Business District (CBD) and one in the suburbs. Geoff is concerned that if he chooses the CBD location, he will have to charge higher prices for his food to cover the higher rent. Geoff is uncertain of the affect that this will have on his annual revenue. To help with his decision, Geoff will carry out a multiple regression analysis to predict the annual revenue earned by restaurants based on their distance from the CBD in miles and the average price in dollars charged for a dish. Geoff has collected a sample of 50 restaurants that are randomly spread around the city. The proximity to the CBD (x1i), average price of a dish (x2i) and annual revenues (yi) were recorded for each of these restaurants (i = 1, 2, ..., 50). Geoff believes that a multiple linear regression is appropriate for this data. After carrying out some analysis, geoff has noticed that moving one mile away from the CBD (assuming that the average price of a dish is held constant) will result in a decrease of $60,000 in annual revenues. Geoff has also noticed that increasing the average price of a dish by $1 (assuming that the distance from the CBD is held constant) will result in an increase of $40,000 in annual revenues. The data shows that a restaurant at the center of the CBD that gave away its food for free would have an annual loss of $20,000. Select the multiple linear regression equation that corresponds to Geoff's analysis: y^i = -20,000 + 40,000x1i - 60,000x2i y^i = 20,000 + 60,000x1i - 40,000x2i y^i = -20,000 - 60,000x1i + 40,000x2i y^i = 20,000 - 40,000x1i + 60,000x2i [2 points]- 5 of 8 ID: MST.MR.TM.03.0040 Select all the scatter plots that indicate a violation of one or more of the assumptions of regression: [1 point]- 6 of 8 ID: MST.MR.TM.03.0020b The following four diagrams depict four residual plots for four different regression models. Select the residual plot that suggests that the assumption of independence of error terms is violated: [3 points]- 7 of 8 ID: MST.MR.TM.04.0010b A multiple regression model is developed by a lecturer in order to study how students fare in their end-ofyear exams. The dependent variable is the score a student achieves in the end-of-year exam (y) and this variable is modeled against two independent variables: their score in the mid-year exam (x 1) and the number of hours they spent studying for the end-of-year exam (x 2). y = 0 + 1x1 + 2x2 + A sample of 60 students is collected. For each student, the student's score in the mid year exam, the number of hours they studied, and their score in the end-of-year exam are recorded. The following regression equation was developed: y^ = 5 + 0.8x1 + 0.4x2 A t test is to be conducted in order to assess the significance of mid-year score in predicting end-of-year score. Select whether the following statements about the test are true or false: True a) The test will assess whether all coefficients except 1 are zero. b) Assuming the null hypothesis is true, the test statistic follows the t distribution with 58 degrees of freedom. c) The test is one-tailed. False PROBLEM 8 Yun has constructed a multiple regression model and has conducted an F test of the model at a level of significance of 0.05. The model has four independent variables. The result of the F test was that the null hypothesis was rejected. Select the appropriate conclusion that can be drawn: the dependent variable is not related to any of the independent variables at least one of the regression coefficients is non-zero all of the regression coefficients are non-zero all of the regression coefficients are zero exactly one of the regression coefficients is non-zero Final score Mid-term Score Hours studying per week Hours watching TV per week 52 55 19 17 73 91 4 25 34 53 2 18 58 69 17 31 49 37 12 17 40 23 19 23 41 28 19 17 47 65 11 22 34 32 9 25 38 33 9 12 59 67 11 18 71 90 12 33 65 84 10 19 28 42 4 12 59 67 15 25 69 75 14 25 72 86 19 21 50 68 7 15 61 54 19 24 28 28 13 8 45 45 19 19 69 77 8 12 52 35 19 6 79 90 20 25 57 84 1 14 58 67 14 19 74 100 16 27 25 23 5 7 38 51 2 32 87 88 19 22