Exam 13: Introduction to 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,what is the p-value for testing whether Wages have a negative impact on corporate sales?

The coefficient of multiple determination is calculated by taking the ratio of the regression sum of squares over the total sum of squares (SSR/SST)and subtracting that value from 1.

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

Instruction 13-16 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,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 13-13 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 null hypothesis H₀: ?₁ = ?₂ = ?₃= 0 implies that percentage of students passing the proficiency test is not related to any of the explanatory variables.

Instruction 13-9 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) X₁ = Length of time in weight-loss program (in months) X₂ = 1 if morning session,0 if not X₃ = 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 = β0 + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₁X₂ + β₅X₁X₃ + ε Partial output from Microsoft Excel follows: Regression Statistics Multiple R 0.73514 R Square 0.540438 Adjusted R Square 0.157469 Standard Error 12.4147 Observations 12 ANOVA F=5.41118 Significance F=0.040201 Intercept Coeff StdError t Stat P -value Length 0.089744 14.127 0.0060 0.9951 Morn Ses 6.22538 2.43473 2.54956 0.0479 Aft Ses 2.217272 22.1416 0.100141 0.9235 Length*Morn Ses 11.8233 3.1545 3.558901 0.0165 Length*Aft Ses 0.77058 3.562 0.216334 0.8359 -0.54147 3.35988 -0.161158 0.8773 -Referring to Instruction 13-9,in terms of the ?'s in the model,give the mean change in weight-loss (Y)for every 1 month increase in time in the program (X₁)when attending the morning session.

Instruction 13-8 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: Regression Analysis Regression Statistics Multiple R 0.63 R Square 0.40 Adjusted R Square 0.23 Standard Error 50.00 Observations 15.00 ANOVA df SS MS F Significance F Regression 3 5994.24 2.40 0.12 Residual 11 27496.82 Total 45479.54 Coefficients Standard t Stat P-value Lower 99.0\% Upper 99.0\% Intercept Error AGE 123.80 48.71 2.54 0.03 -27.47 275.07 TICKETS -0.82 0.87 -0.95 0.36 -3.51 1.87 DENSITY 21.25 10.66 1.99 0.07 -11.86 54.37 -3.14 6.46 -0.49 0.64 -23.19 16.91 -Referring to Instruction 13-8,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 13-5 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,what is the p-value for testing whether Wages have a positive impact on corporate sales?

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 decided to construct a 95% confidence interval for β₁. The confidence interval is from ________ to ________.

In trying to obtain a model to estimate grades on a statistics test,a lecturer wanted to include,among other factors,whether the person had taken the course previously.To do this,the lecturer included a dummy variable in her regression that was equal to 1 if the person had previously taken the course,and 0 otherwise.The interpretation of the coefficient associated with this dummy variable would be the mean amount the repeat students tended to be above or below non-repeaters,with all other factors the same.

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 for the significance of the coefficient on aggregate price index,the p-value is

A regression had the following results: SST = 102.55,SSE = 82.04.It can be said that 90.0% 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,you can conclude that mean teacher salary has no impact on mean percentage of students passing the proficiency test at a 5% level of significance using the 95% confidence interval estimate for ?₂.

Instruction 13-7 You decide to predict petrol prices in different cities and towns in Australia for your term project.Your dependent variable is price of petrol per litre and your explanatory variables are per capita income,the number of firms that manufacture automobile parts in and around the city,the number of new business starts in the last year,population density of the city,percentage of local taxes on petrol,and the number of people using public transportation.You collected data of 32 cities and obtained a regression sum of squares SSR = 122.8821.Your computed value of standard error of the estimate is 1.9549. -Referring to Instruction 13-7,the value of adjusted r² 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,the null hypothesis should be rejected at a 10% level of significance when testing whether age has any effect on the number of weeks a worker is unemployed due to a layoff.

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,there is sufficient evidence that 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 at a 10% level of significance.