Exam 13: Introduction to Multiple Regression

Instruction 13.34 An automotive engineer would like to be able to predict automobile fuel economy. She believes that the two most important characteristics that affect economy are engine power and the number of cylinders (4 or 6) of a car. She believes that the appropriate model is Y=40-0.05+20-0.1 where = engine power =1 if 4 cylinders, 0 if 6 cylinders Y = economy expressed as kilometres. -Referring to Instruction 13.34,the predicted number of kilometres for a 300 engine power,6-cylinder car is ___________.

A multiple regression is called 'multiple' because it has several explanatory variables.

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,____ of the variation in the number of weeks a worker is unemployed due to a layoff can be explained by the six independent variables after taking into consideration the number of independent variables and the number of observations.

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,which of the following is a correct statement?

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,which of the following is the correct null hypothesis to test whether instructional spending per pupil has any effect on percentage of students passing the proficiency test?

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 null hypothesis H₀: β₁ = β₂ = β₃= 0 implies that percentage of students passing the proficiency test is not related to any of the explanatory variables.

The coefficient of multiple determination r² measures the proportion of variation in Y that is explained by X₁ and X₂.

You have just computed a regression in which the value of coefficient of multiple determination is 0.57.To determine if this indicates that the independent variables explain a significant portion of the variation in the dependent variable,you would perform an F test.

Instruction 13.17 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 13.17,when the microeconomist used a simple linear regression model with sales as the dependent variable and wages as the independent variable,he obtained an r² value of 0.601.What additional percentage of the total variation of sales has been explained by including capital spending in the multiple regression?

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 value of the adjusted coefficient of multiple determination,adjusted r²,is ___________.

When an explanatory variable is dropped from a multiple regression model,the coefficient of multiple determination can increase.

Instruction 13.36 The Head of the Accounting Department wanted to see if she could predict the average grade of students using the number of course units (credits) and total university entrance exam scores of each. She takes a sample of students and generates the following Microsoft Excel output: OUTPUT SUMMARY Regression Statistics MultipleR 0.916 R Square 0.839 Adj. R Square 0.732 Std. Error 0.24685 Observations 6 ANOVA df SS MS F Signif F Regression 2 0.95219 0.47610 7.813 0.0646 Residual 3 0.18281 0.06094 Total 5 1.13500 Coeff StdError t Stat p value Intercept 4.593897 1.13374542 4.052 0.0271 GDP -0.247270 0.06268485 -3.945 0.0290 Price 0.001443 0.00101241 1.425 0.2494 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.36,the Head of Department wants to use a t test to test for the significance of the coefficient of X₁.At a level of significance of 0.05,the department head would decide that ?₁ ? 0.

Variance inflationary factor (VIF)states the degree to which collinearity among the predictors increases the precision of an estimate.

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,you can conclude that,holding constant the effect of the other independent variables,there is a difference in the mean number of weeks a worker is unemployed due to a layoff between a worker who is married and one who is not at a 10% level of significance if you use only the information of the 95% confidence interval estimate for ?₄.

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 related to all of the explanatory variables.

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 p-value of the test statistic when testing 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 13.35 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,what is the value of the test statistics when testing whether instructional spending per pupil has any effect on percentage of students passing the proficiency test,taking into account the effect of all the other independent 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,there is sufficient evidence that the percentage of students passing the proficiency test depends on all of the explanatory variables.

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,suppose the builder wants to test whether the coefficient on School is significantly different from 0.What is the value of the relevant t-statistic?

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,you can conclude that,holding constant the effect of the other independent variables,there is a difference in the mean number of weeks a worker is unemployed due to a layoff between a worker who is married and one who is not at a 5% level of significance if you use only the information of the 95% confidence interval estimate for ?₄.