Exam 13: Introduction to Multiple Regression

A regression had the following results: SST = 82.55,SSE = 29.85.It can be said that 73.4% of the variation in the dependent variable is explained by the independent variables in the regression.

Instruction 13-13 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 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 ANOVA df SS MS F Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value 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 13-13,the alternative hypothesis H₁: At least one of ?_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 13-15 A financial analyst wanted to examine the relationship between salary (in $1,000)and 4 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: SUMMARY Regression Statistics Multiple R 0.992 R Square 0.984 Adj. R Square 0.979 Std. Error 2.26743 Observations 20 ANOVA SS MS Signif Regression 4 4609.83164 1152.45791 224.160 0.0001 Residual 15 77.11836 5.14122 Total 19 4686.95000 Coeff StdError Stat -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 13-15,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 13-4 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 Siguif Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff SttError Stat -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 13-4,when the builder used a simple linear regression model with house size (House)as the dependent variable and education (School)as the independent variable,he obtained an r² value of 23.0%.What additional percentage of the total variation in house size has been explained by including family size and income in the multiple regression?

Instruction 13-5 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. SUMMARY Regression Statistics \begin{tabular} { l r } Multiple $\mathrm { R }$ & $0.830$ \\ \hline Square & $0.689$ \end{tabular} R Square $\quad 0.689$ Adj. R Square $\quad 0.662$ Std. Error $\quad 17501.643$ Observations 26 ANOVA SS MS Siguif Regression 2 15579777040 7789888520 25.432 0.0001 Residual 23 7045072780 306307512 Total 25 22624849820 Coeff StdError Stat -value Intercept 15800.0000 6038.2999 2.617 0.0154 Capital 0.1245 0.2045 0.609 0.5485 Wages 7.0762 1.4729 4.804 0.0001 Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 13-5,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 13-16 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. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error Observations 40 ANOVA df SS MS F Significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% 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: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 ANOVA df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 Coefficients Standard Error 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 13-16 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 1% level of significance if all you have is the information of the 95% confidence interval estimate for ?₄.

Instruction 13-3 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. SUMMARY Regression Statistics Multiple R 0.991 R Square 0.982 Adj. R Square 0.976 Std. Error 0.299 Observations 10 ANOVA SS MS Signiff Regression 2 33.4163 16.7082 186.325 0.0001 Residual 7 0.6277 0.0897 Total 9 34.0440 Coeff StaError Stat -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 13-3,to test whether aggregate price index has a negative impact on consumption,the p-value is

Instruction 13-16 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. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error Observations 40 ANOVA df SS MS F Significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P -value Lower 95\% Upper 95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 7.8257 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: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 ANOVA df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 Coefficients Standard Error 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 13-16 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 13-3 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. SUMMARY Regression Statistics Multiple R 0.991 R Square 0.982 Adj. R Square 0.976 Std. Error 0.299 Observations 10 ANOVA SS MS Signiff Regression 2 33.4163 16.7082 186.325 0.0001 Residual 7 0.6277 0.0897 Total 9 34.0440 Coeff StaError Stat -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 13-3,the p-value for GDP is

Instruction 13-3 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. $\quad\quad\quad\quad\quad$ OUTPUT SUMMARY Regression $\quad$ Statistics Multiple R 0.991 R Square 0.982 Adj. R Square 0.976 Std. Error 0.299 Observations 10 ANOVA SS MS SignifF Regression 2 33.4163 16.7082 186.325 0.0001 Residual 7 0.6277 0.0897 Total 9 34.0440 Coeff StaError Stat -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 13-3,the p-value for the regression model as a whole is

Instruction 13-3 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. $\quad\quad\quad\quad\quad$ OUTPUT SUMMARY Regression $\quad$ Statistics Multiple R 0.991 R Square 0.982 Adj. R Square 0.976 Std. Error 0.299 Observations 10 ANOVA SS MS SignifF Regression 2 33.4163 16.7082 186.325 0.0001 Residual 7 0.6277 0.0897 Total 9 34.0440 Coeff StaError Stat -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 13-3,what is the estimated average consumption level for an economy with GDP equal to $4 billion and an aggregate price index of 150?

Instruction 13-10 As a project for his business statistics class,a student examined the factors that determined parking meter rates throughout the campus area.Data were collected for the price per hour of parking,number of city blocks to the centre of the university,and one of the three jurisdictions: on campus,in the CBD and off campus,or outside of the CBD and off campus.The population regression model hypothesised is: Yi = α + β₁x_1i + β₂x_2i + β₃x_2i + ε_i Where Y is the meter price x₁ is the number of blocks to the centre of the university x₂ is a dummy variable that takes the value 1 if the meter is located in the CBD and off campus and the value 0 otherwise x₃ is a dummy variable that takes the value 1 if the meter is located outside of the CBD and off campus,and the value 0 otherwise The following Excel results are obtained. Regression Statistics Multiple R 0.9659 R Square 0.9331 Adjusted R Square 0.9294 Standard Error 0.0327 Observations 58 ANOVA Df SS MS SS Significance F Regression 3 0.8094 0.2698 251.1995 0.0000 Residual 54 0.0580 0.0010 Total 57 0.8675 Coefficients Standard Error t Stat P -value Intercept 0.5118 0.0136 37.4675 2.4904 -0.0045 0.0034 -1.3276 0.1898 -0.2392 0.0123 -19.3942 0.0000 -0.0002 0.0123 -0.0214 0.9829 -Referring to Instruction 13-10,predict the meter rate per hour if one parks outside of the CBD and off campus three blocks from the centre of the university.

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: $\begin{array}{clll}\underline{\text { Employee }} & \underline{Y(\$)} & \underline{X}_{1} &\underline{X}_{2}\\1 & 10 & 3 & 0 \\2 & 12 & 1 & 5 \\3 & 15 & 8 & 1 \\4 & 17 & 5 & 8 \\5 & 20 & 7 & 12 \\6 & 25 & 10 & 9\end{array}$ -Referring to Instruction 13-2,for these data,what is the estimated coefficient for performance rating,b₁?

Instruction 13-14 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: SUMMARY Regression Statistics Multiple R 0.916 R Square 0.839 Adj. R Square 0.732 Std. Error 0.24685 Observations 6 ANOVA SS MS Signif Regression 2 0.95219 0.47610 7.813 0.0646 Residual 3 0.18281 0.06094 Total 5 1.13500 Coeff StdError Stat -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 13-14,the Head of Department wants to test H₀: β₁ = β₂ = 0.The value of the F test statistic is ________.

Instruction 13-14 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: SUMMARY Regression Statistics Multiple R 0.916 R Square 0.839 Adj. R Square 0.732 Std. Error 0.24685 Observations 6 ANOVA SS MS Signif Regression 2 0.95219 0.47610 7.813 0.0646 Residual 3 0.18281 0.06094 Total 5 1.13500 Coeff StdError Stat -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 13-14,the estimate of the unit change in the mean of Y per unit change in X₁,holding X₂ constant,is ________.

Instruction 13-16 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. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error Observations 40 ANOVA df SS MS F Significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P -value Lower 95\% Upper 95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 7.8257 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: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 ANOVA df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 Coefficients Standard Error 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 13-16 Model 1,what is the p-value of 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 13-5 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. SUMMARY Regression Statistics \begin{tabular} { l r } Multiple $\mathrm { R }$ & $0.830$ \\ \hline Square & $0.689$ \end{tabular} R Square $\quad 0.689$ Adj. R Square $\quad 0.662$ Std. Error $\quad 17501.643$ Observations 26 ANOVA SS MS Siguif Regression 2 15579777040 7789888520 25.432 0.0001 Residual 23 7045072780 306307512 Total 25 22624849820 Coeff StdError Stat -value Intercept 15800.0000 6038.2999 2.617 0.0154 Capital 0.1245 0.2045 0.609 0.5485 Wages 7.0762 1.4729 4.804 0.0001 Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 13-5,suppose the microeconomist wants to test whether the coefficient on Capital is significantly different from 0.What is the value of the relevant t-statistic?

When an explanatory variable is dropped from a multiple regression model,the adjusted r² can increase.

Instruction 13-13 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 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 ANOVA df SS MS F Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value 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 13-13,what is the value of the test statistics when testing whether daily mean of the percentage of students attending class has any effect on percentage of students passing the proficiency test,taking into account the effect of all the other independent variables?

Instruction 13-13 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 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 ANOVA df SS MS F Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value 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 13-13,the alternative hypothesis H₁: At least one of ?_j ? 0 for j = 1,2,3 implies that percentage of students passing the proficiency test is affected by at least one of the explanatory variables.