Exam 13: Introduction to Multiple Regression

Instruction 13.25 Given below are results from the regression analysis where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Unemploy) and the independent variables are the age of the worker (Age), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. Model 1 Regression Statistics Multiple R 0.7035 R Square 0.4949 Adj. R Square 0.4030 Std. Error 18.4861 Observations 40 ANOVA df SS MS F Signiff Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 223325.9 Coeff StdError tStat p value Lower 95\% Upper95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Mode 2 Regression Statistics Multiple R 0.6391 R Square 0.4085 Adj. R Square 0.3765 Std. Error 18.8929 Observations 40 ANOVA df SS MS F Signiff Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 Coeff StdError t Stat p value Intercept -0.2143 11.5796 -0.0185 0.9853 Age 1.4448 0.3160 4.5717 0.0000 Manager -22.5761 11.3488 -1.9893 0.0541 -Referring to Instruction 13.25 Model 1,the null hypothesis should be rejected at a 10% level of significance when testing whether there is a significant relationship between the number of weeks a worker is unemployed due to a layoff and the entire set of explanatory variables.

Instruction 13.22 The education department's regional executive officer wanted to predict the percentage of students passing a Grade 6 proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries) and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: -Referring to Instruction 13.22,which of the following is a correct statement?

Instruction 13.27 A real estate builder wishes to determine how house size (House) is influenced by family income (Income), family size (Size) and education of the head of household (School). House size is measured in hundreds of square metres, income is measured in thousands of dollars and education is in years. The builder randomly selected 50 families and ran the multiple regression. Microsoft Excel output is provided below: OUTPUT SUMMARY Regression Statistics Multiple R 0.865 R Square 0.748 Adj. R Square 0.726 Std. Error 5.195 Observations 50 ANOVA df SS MS F Signiff Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff StdError t Stat p value Intercept -1.6335 5.8078 -0.281 0.7798 Income 0.4485 0.1137 3.9545 0.0003 Size 4.2615 0.8062 5.286 0.0001 School -0.6517 0.4319 -1.509 0.1383 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.27,one individual in the sample had an annual income of $100,000,a family size of 10 and an education of 16 years.This individual owned a home with an area of 7,000 square metre (House = 70.00).What is the residual (in hundreds of square metre)for this data point?

Instruction 13.28 A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies. She proceeds to randomly select 26 large corporations and record information in millions of dollars. The Microsoft Excel output below shows results of this multiple regression. Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.28,one company in the sample had sales of $20 billion (Sales = 20,000).This company spent $300 million on capital and $700 million on wages.What is the residual (in millions of dollars)for this data point?

Instruction 13.35 The education department's regional executive officer wanted to predict the percentage of students passing a Grade 6 proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries) and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: -Referring to Instruction 13.35,there is sufficient evidence that instructional spending per pupil has an effect on percentage of students passing the proficiency test,while holding constant the effect of all the other independent variables at a 5% level of significance.

The total sum of squares (SST)in a regression model will never exceed the regression sum of squares (SSR).

Instruction 13.24 A financial analyst wanted to examine the relationship between salary (in $1,000) and four variables: age (X₁ = Age), experience in the field (X₂ = Exper), number of degrees (X₃ = Degrees) and number of previous jobs in the field (X₄ = Prevjobs). He took a sample of 20 employees and obtained the following Microsoft Excel output: Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.24,the analyst wants to use a t test to test for the significance of the coefficient of X₃.For a level of significance of 0.01,the critical values of the test are _____.

Instruction 13.2 A lecturer in industrial relations believes that an individual's wage rate at a factory (Y) depends on his performance rating (X₁) and the number of economics courses the employee successfully completed at university (X₂). The lecturer randomly selects six workers and collects the following information: Employee Y(\ ) X1 X2 1 10 3 0 2 12 1 5 3 15 8 1 4 17 5 8 5 20 7 12 6 25 10 9 -Referring to Instruction 13.2,an employee who took 12 economics courses scores 10 on the performance rating.What is her estimated expected wage rate?

Instruction 13.30 A real estate builder wishes to determine how house size (House) is influenced by family income (Income), family size (Size) and education of the head of household (School). House size is measured in hundreds of square metres, income is measured in thousands of dollars and education is in years. The builder randomly selected 50 families and ran the multiple regression. Microsoft Excel output is provided below: OUTPUT SUMMARY Regression Statistics Multiple R 0.865 R Square 0.748 Adj. R Square 0.726 Std. Error 5.195 Observations 50 ANOVA df SS MS F Signiff Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff StdError t Stat p value Intercept -1.6335 5.8078 -0.281 0.7798 Income 0.4485 0.1137 3.9545 0.0003 Size 4.2615 0.8062 5.286 0.0001 School -0.6517 0.4319 -1.509 0.1383 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.30,at the 0.01 level of significance,what conclusion should the builder draw regarding the inclusion of School in the regression model?

Instruction 13.14 Given below are results from the regression analysis where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Unemploy) and the independent variables are the age of the worker (Age), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. Model 1 Regression Statistics Multiple R 0.7035 R Square 0.4949 Adj. R Square 0.4030 Std. Error 18.4861 Observations 40 ANOVA df SS MS F Signiff Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 223325.9 Coeff StdError tStat p value Lower 95\% Upper95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Mode 2 Regression Statistics Multiple R 0.6391 R Square 0.4085 Adj. R Square 0.3765 Std. Error 18.8929 Observations 40 ANOVA df SS MS F Signiff Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 Coeff StdError t Stat p value Intercept -0.2143 11.5796 -0.0185 0.9853 Age 1.4448 0.3160 4.5717 0.0000 Manager -22.5761 11.3488 -1.9893 0.0541 -Referring to Instruction 13.14 Model 1,predict the number of weeks being unemployed due to a layoff for a worker who is a 30-year-old,has 10 years of education,has 15 years of experience at the previous job,is married,is the head of household and is a manager.

Instruction 13.18 One of the most common questions of prospective house buyers pertains to the average cost of heating in dollars (Y). To provide its customers with information on that matter, a large real estate firm used the following four variables to predict heating costs: the daily minimum outside temperature in degrees of Celsius (X₁), the amount of insulation in cm (X₂), the number of windows in the house (X₃) and the age of the furnace in years (X₄). Given below are the Microsoft Excel outputs of two regression models. -Referring to Instruction 13.18,what is the value of the partial F test statistic for H₀: β₃ = β₄ = 0 vs.H₁: At least one β_j ≠ 0,j = 3,4?

Instruction 13.37 Given below are results from the regression analysis where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Unemploy) and the independent variables are the age of the worker (Age), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. Model 1 Regression Statistics Multiple R 0.7035 R Square 0.4949 Adj. R Square 0.4030 Std. Error 18.4861 Observations 40 ANOVA df SS MS F Signif F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 223325.9 Coeff StdError tStat p value Lower 95\% Upper95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Mode 2 Regression Statistics Multiple R 0.6391 R Square 0.4085 Adj. R Square 0.3765 Std. Error 18.8929 Observations 40 ANOVA df SS MS F Signif F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 Coeff StdError t Stat p value Intercept -0.2143 11.5796 -0.0185 0.9853 Age 1.4448 0.3160 4.5717 0.0000 Manager -22.5761 11.3488 -1.9893 0.0541 -Referring to Instruction 13.37 Model 1,what are the numerator and denominator degrees of freedom,respectively,for the test statistic to determine whether there is a significant relationship between the number of weeks a worker is unemployed due to a layoff and the entire set of explanatory variables?

Instruction 13.20 You worked as an intern at We Always Win Car Insurance Company last summer. You notice that individual car insurance premium depends very much on the age of the individual, the number of traffic tickets received by the individual and the population density of the city in which the individual lives. You performed a regression analysis in Microsoft Excel and obtained the following information: -Referring to Instruction 13.20,the F test for the significance of the entire regression performed at a level of significance of 0.01 leads to a rejection of the null hypothesis.

Instruction 13.22 The education department's regional executive officer wanted to predict the percentage of students passing a Grade 6 proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries) and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: -Referring to Instruction 13.22,what is the value of the test statistic to determine whether there is a significant relationship between percentage of students passing the proficiency test and the entire set of explanatory variables?

Instruction 13.38 A weight-loss clinic wants to use regression analysis to build a model for weight-loss of a client (measured in kilograms). Two variables thought to effect weight-loss are client's length of time on the weight loss program and time of session. These variables are described below: $Y =$ Weight-loss (in kilograms) $X _ { 1 } =$ Length of time in weight-loss program (in months) $x _ { 2 } = 1$ if morning session, 0 if not $x _ { 3 } = 1$ if afternoon session, 0 if not (Base level = evening session) Data for 12 clients on a weight-loss program at the clinic were collected and used to fit the interaction model: $Y = \beta 0 + \beta _ { 1 } X _ { 1 } + \beta _ { 2 } X _ { 2 } + \beta _ { 3 } X _ { 3 } + \beta _ { 4 } X _ { 1 } X _ { 2 } + \beta _ { 5 } X _ { 1 } X _ { 3 } + \varepsilon$ Partial output from Microsoft Excel follows: -Referring to Instruction 13.38,which of the following statements is supported by the analysis shown?

Instruction 13.25 Given below are results from the regression analysis where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Unemploy) and the independent variables are the age of the worker (Age), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. Model 1 Regression Statistics Multiple R 0.7035 R Square 0.4949 Adj. R Square 0.4030 Std. Error 18.4861 Observations 40 ANOVA df SS MS F Signiff Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 223325.9 Coeff StdError tStat p value Lower 95\% Upper95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Mode 2 Regression Statistics Multiple R 0.6391 R Square 0.4085 Adj. R Square 0.3765 Std. Error 18.8929 Observations 40 ANOVA df SS MS F Signiff Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 Coeff StdError t Stat p value Intercept -0.2143 11.5796 -0.0185 0.9853 Age 1.4448 0.3160 4.5717 0.0000 Manager -22.5761 11.3488 -1.9893 0.0541 -Referring to Instruction 13.25 Model 1,there is sufficient evidence that the number of weeks a worker is unemployed due to a layoff depends on at least one of the explanatory variables at a 10% level of significance.

Instruction 13.29 An economist is interested to see how consumption for an economy (in $ billions) is influenced by gross domestic product ($ billions) and aggregate price (consumer price index). The Microsoft Excel output of this regression is partially reproduced below. OUTPUT SUMMARY Regression Statistics MultipleR 0.991 R Square 0.982 Adj. R Square 0.976 Std. Error 0.299 Observations 10 ANOVA df SS MS F Signiff Regression 2 33.4163 16.7082 186.325 0.0001 Residual 7 0.6277 0.0897 Total 9 34.0440 Coeff StdError t Stat p value Intercept -1.6335 0.5674 -0.152 0.8837 GDP 0.7654 0.0574 13.340 0.0001 Price -0.0006 0.0028 -0.219 0.8330 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.29,to test for the significance of the coefficient on aggregate price index,the value of the relevant t-statistic is

Instruction 13.37 Given below are results from the regression analysis where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Unemploy) and the independent variables are the age of the worker (Age), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. Model 1 Regression Statistics Multiple R 0.7035 R Square 0.4949 Adj. R Square 0.4030 Std. Error 18.4861 Observations 40 ANOVA df SS MS F Signif F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 223325.9 Coeff StdError tStat p value Lower 95\% Upper95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Mode 2 Regression Statistics Multiple R 0.6391 R Square 0.4085 Adj. R Square 0.3765 Std. Error 18.8929 Observations 40 ANOVA df SS MS F Signif F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 Coeff StdError t Stat p value Intercept -0.2143 11.5796 -0.0185 0.9853 Age 1.4448 0.3160 4.5717 0.0000 Manager -22.5761 11.3488 -1.9893 0.0541 -Referring to Instruction 13.37 Model 1,what is the value of the test statistic when testing whether age has any effect on the number of weeks a worker is unemployed due to a layoff,while holding constant the effect of all the other independent variables?

Instruction 13.35 The education department's regional executive officer wanted to predict the percentage of students passing a Grade 6 proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries) and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: -Referring to Instruction 13.35,there is sufficient evidence that daily mean of the percentage of students attending class has an effect on percentage of students passing the proficiency test,while holding constant the effect of all the other independent variables at a 5% level of significance.

Instruction 13.20 You worked as an intern at We Always Win Car Insurance Company last summer. You notice that individual car insurance premium depends very much on the age of the individual, the number of traffic tickets received by the individual and the population density of the city in which the individual lives. You performed a regression analysis in Microsoft Excel and obtained the following information: -Referring to Instruction 13.20,to test the significance of the multiple regression model,what is the form of the null hypothesis?