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 alternative hypothesis $H _ { 1 }$ : At least one of ?_j ? 0 for j = 1,2,3,4,5,6 implies that the number of weeks a worker is unemployed due to a layoff is related to at least one of the explanatory variables.

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 H₀: ?₁ = ?₂ = ?₃ = ?₄ = ?₅ = ?₆ = 0 implies that the number of weeks a worker is unemployed due to a layoff is not affected by any of the explanatory variables.

From the coefficient of multiple determination,you cannot detect the strength of the relationship between Y and any individual independent variable.

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 and allowing for a 1% probability of committing a Type I error,what is the decision and conclusion for the test H₀: β₁ = β₂ = β₃ = β₄ = 0 vs.H₁: At least one β_j ≠ 0,j = 1,2,...,4 using Model 1?

Instruction 13.13 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: $\text { OUTPUT }$ SUMMARY Regression Statistics Multiple R 0.992 R Square 0.984 Adj. R Square 0.979 Std. Error 2.26743 Observations 20 ANOVA df SS MS F Signif F Regression 4 4609.83164 1152.45791 224.160 0.0001 Residual 15 77.11836 5.14122 Total 19 4686.95000 Coeff Std Error t Stat p value Intercept -9.611198 2.77988638 -3.457 0.0035 Age 1.327695 0.11491930 11.553 0.0001 Exper -0.106705 0.14265559 -0.748 0.4660 Degrees 7.311332 0.80324187 9.102 0.0001 Prevjobs -0.504168 0.44771573 -1.126 0.2778 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.13,the critical value of an F test on the entire regression for a level of significance of 0.01 is ___________.

Instruction 13.33 An econometrician is interested in evaluating the relation of demand for building materials to mortgage rates in Sydney and Melbourne. He believes that the appropriate model is Y=10+5+8 where = mortgage rate in \% =1 if Sydney, 0 if Melbourne Y = demand in \ 100 per capita -Referring to Instruction 13.33,the predicted demand in Melbourne when the mortgage rate is 8% is ___________ per capita.

The denominator degrees of freedom for the partial F statistic are

You need to proceed with extreme caution when using a multiple regression model that has one or more large VIF values.

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,which of the following is the correct alternative hypothesis to test whether being married or not makes a difference in the mean number of weeks a worker is unemployed due to a layoff,while holding constant the effect of all the other independent variables?

Instruction 13.15 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 13.15,when the economist used a simple linear regression model with consumption as the dependent variable and GDP as the independent variable,he obtained an r² value of 0.971.What additional percentage of the total variation of consumption has been explained by including aggregate prices in the multiple regression?

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 p-value is

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,there is sufficient evidence that at least one of the explanatory variables is related to the percentage of students passing the proficiency test.

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,estimate the mean 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 13.10 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: Regression Statistics Multiple R 0.7930 R Square 0.6288 Adj. R Square 0.602 Standard Error 10.4570 Observations 47 ANOVA d SS MS F Signif F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coeff StolFrro tSta p-va/ue Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 0.0000 -957.3401 -549.5050 \% Attendance 8.5014 1.0771 7.8929 0.0000 6.3292 10.6735 Salary 0.000000685 0.0006 0.0011 0.9991 -0.0013 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 -Referring to Instruction 13.10,estimate the mean percentage of students passing the proficiency test for all the schools that have a daily mean of 95% of students attending class,a mean teacher salary of 40,000 dollars and an instructional spending per pupil of 2,000 dollars.

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 decided to construct a 99% confidence interval for $\beta$ ₃.The confidence interval is from _____ to _____.

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,the alternative hypothesis H₁: At least one of β_j ≠ 0 for j = 1,2,3 implies that percentage of students passing the proficiency test is affected by all of the explanatory variables.

Instruction 13.13 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: $\text { OUTPUT }$ SUMMARY Regression Statistics Multiple R 0.992 R Square 0.984 Adj. R Square 0.979 Std. Error 2.26743 Observations 20 ANOVA df SS MS F Signif F Regression 4 4609.83164 1152.45791 224.160 0.0001 Residual 15 77.11836 5.14122 Total 19 4686.95000 Coeff Std Error t Stat p value Intercept -9.611198 2.77988638 -3.457 0.0035 Age 1.327695 0.11491930 11.553 0.0001 Exper -0.106705 0.14265559 -0.748 0.4660 Degrees 7.311332 0.80324187 9.102 0.0001 Prevjobs -0.504168 0.44771573 -1.126 0.2778 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.13,the p-value of the F test for the significance of the entire regression is ___________.

Instruction 13.16 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 13.16,which of the following values for the level of significance is the smallest for which all explanatory variables are significant individually?

In a multiple regression model,the value of the coefficient of multiple determination

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 the correct alternative hypothesis to determine whether there is a significant relationship between percentage of students passing the proficiency test and the entire set of explanatory variables?