Exam 13: Multiple Regression

SCENARIO 13-9 You decide to predict gasoline prices in different cities and towns in the United States for your term project.Your dependent variable is price of gasoline per gallon and your explanatory variables are per capita income and the number of firms that manufacture automobile parts in and around the city.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 SCENARIO 13-9, the value of adjusted r 2 is

SCENARIO 13-7 The department head of the accounting department wanted to see if she could predict the GPA of students using the number of course units and total SAT scores of each.She takes a sample of 6 students and generates the following Microsoft Excel output: SUMMARY OUTPUT Regression Statistics Multiple R 0.916 R Square 0.839 Adjusted R Square 0.732 Standard 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 Units -0.247270 0.06268485 -3.945 0.0290 Total SAT 0.001443 0.00101241 1.425 0.2494 -Referring to SCENARIO 13-7, the department head wants to use a t test to test for the significance of the coefficient of X1.For a level of significance of 0.05, the critical values of the test are .

SCENARIO 13-17 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) and a dummy variable for management position (Manager: 1 = yes, 0 = no). The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R Square 0.3765 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 SCENARIO 13-17, which of the following is the correct alternative hypothesis to test whether age has any effect on the number of weeks a worker is unemployed due to a layoff while holding constant the effect of the other independent variable?

SCENARIO 13-4 A real estate builder wishes to determine how house size (House) is influenced by family income (Income) and family size (Size).House size is measured in hundreds of square feet and income is measured in thousands of dollars.The builder randomly selected 50 families and ran the multiple regression.Partial Microsoft Excel output is provided below: Regression Statistics Multiple R 0.8479 R Square 0.7189 Adjusted R Square 0.7069 Standard Error 17.5571 Observations 50 $\text { ANOVA }$ df SS MS F Significance F Regression 37043.3236 18521.6618 0.0000 Residual 14487.7627 308.2503 Total 49 51531.0863 Coefficients Standard Error t Stat P-value Intercept -5.5146 7.2273 -0.7630 0.4493 Income 0.4262 0.0392 10.8668 0.0000 Size 5.5437 1.6949 3.2708 0.0020 $\text { Also } \operatorname{SSR}\left(X_{1} \mid X_{2}\right)=36400.6326 \text { and } \operatorname{SSR}\left(X_{2} \mid X_{1}\right)=3297.7917$ -Referring to SCENARIO 13-4, suppose the builder wants to test whether the coefficient onIncome is significantly different from 0.What is the value of the relevant t-statistic?

SCENARIO 13-8 A financial analyst wanted to examine the relationship between salary (in $1,000) and 2 variables: age (X1 = Age) and experience in the field (X2 = Exper).He took a sample of 20 employees and obtained the following Microsoft Excel output: Regression Statistics Multiple R 0.8535 R Square 0.7284 Adjusted R Square 0.6964 Standard Error 10.5630 Observations 20 ANOYA df SS MS F Siqnificonce F Regression 2 5086.5764 2543.2882 22.7941 0.0000 Residual 17 1896.8050 111.5768 Total 19 6983.3814 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 1.5740 9.2723 0.1698 0.8672 -17.9888 21.1368 Age 1.3045 0.1956 6.6678 0.0000 0.8917 1.7173 Exper -0.1478 0.1944 -0.7604 0.4574 -0.5580 0.2624 -Referring to SCENARIO 13-8, the analyst wants to use a t test to test for the significance of the coefficient of X2.At a level of significance of 0.01, the department head would decide that $\beta$ 2 $\neq$ 0 .

SCENARIO 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. -Referring to SCENARIO 13-3, to test for the significance of the coefficient on aggregate price index, the p-value is

SCENARIO 13-8 A financial analyst wanted to examine the relationship between salary (in $1,000) and 2 variables: age (X1 = Age) and experience in the field (X2 = Exper).He took a sample of 20 employees and obtained the following Microsoft Excel output: Regression Statistics Multiple R 0.8535 R Square 0.7284 Adjusted R Square 0.6964 Standard Error 10.5630 Observations 20 ANOYA df SS MS F Siqnificonce F Regression 2 5086.5764 2543.2882 22.7941 0.0000 Residual 17 1896.8050 111.5768 Total 19 6983.3814 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 1.5740 9.2723 0.1698 0.8672 -17.9888 21.1368 Age 1.3045 0.1956 6.6678 0.0000 0.8917 1.7173 Exper -0.1478 0.1944 -0.7604 0.4574 -0.5580 0.2624 -Referring to SCENARIO 13-8, the value of the partial F test statistic is forH0: Variable X1 does not significantly improve the model after variable X2 has been includedH1: Variable X1 significantly improves the model after variable X2 has been included

To explain personal consumption (CONS) measured in dollars, data is collected forINC: personal income in dollarsCRDTLIM: $1 plus the credit limit in dollars available to the individualAPR: mean annualized percentage interest rate for borrowing for the individualADVT: per person advertising expenditure in dollars by manufacturers in the city where the individual livesSEX: gender of the individual; 1 if female, 0 if maleA regression analysis was performed with CONS as the dependent variable and CRDTLIM, APR, ADVT, and SEX as the independent variables.The estimated model wasY = 2.28 - 0.29 CRDTLIM + 5.77 APR + 2.35 ADVT +0.39 SEXWhat is the correct interpretation for the estimated coefficient for SEX?

When an additional explanatory variable is introduced into a multiple regression model, the adjusted r 2 can never decrease.

SCENARIO 13-4 A real estate builder wishes to determine how house size (House) is influenced by family income (Income) and family size (Size).House size is measured in hundreds of square feet and income is measured in thousands of dollars.The builder randomly selected 50 families and ran the multiple regression.Partial Microsoft Excel output is provided below: Regression Statistics Multiple R 0.8479 R Square 0.7189 Adjusted R Square 0.7069 Standard Error 17.5571 Observations 50 $\text { ANOVA }$ df SS MS F Significance F Regression 37043.3236 18521.6618 0.0000 Residual 14487.7627 308.2503 Total 49 51531.0863 Coefficients Standard Error t Stat P-value Intercept -5.5146 7.2273 -0.7630 0.4493 Income 0.4262 0.0392 10.8668 0.0000 Size 5.5437 1.6949 3.2708 0.0020 $\text { Also } \operatorname{SSR}\left(X_{1} \mid X_{2}\right)=36400.6326 \text { and } \operatorname{SSR}\left(X_{2} \mid X_{1}\right)=3297.7917$ -Referring to SCENARIO 13-4, which of the following values for the level of significance is the smallest for which each explanatory variable is significant individually?

SCENARIO 13-19 The marketing manager for a nationally franchised lawn service company would like to study the characteristics that differentiate home owners who do and do not have a lawn service.A random sample of 30 home owners located in a suburban area near a large city was selected; 11 did not have a lawn service (code 0) and 19 had a lawn service (code 1).Additional information available concerning these 30 home owners includes family income (Income, in thousands of dollars) and lawn size (Lawn Size, in thousands of square feet). The PHStat output is given below: Binary Logistic Regression Predictor Coefficients SE Coef Z p -Value Intercept -7.8562 3.8224 -2.0553 0.0398 Income 0.0304 0.0133 2.2897 0.0220 Lawn Size 1.2804 0.6971 1.8368 0.0662 Deviance 25.3089 -Referring to SCENARIO 13-19, what is the estimated odds ratio for a home owner with a family income of $50,000 and a lawn size of 2,000 square feet?

SCENARIO 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. -Referring to SCENARIO 13-3, to test whether aggregate price index has a positive impact on consumption, the p-value is

SCENARIO 13-17 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) and a dummy variable for management position (Manager: 1 = yes, 0 = no). The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R Square 0.3765 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 SCENARIO 13-17, 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.

SCENARIO 13-17 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) and a dummy variable for management position (Manager: 1 = yes, 0 = no). The results of the regression analysis are given below: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R Square 0.3765 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 SCENARIO 13-17, 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?

SCENARIO 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. -Referring to SCENARIO 13-3, to test whether gross domestic product has a positive impact on consumption, the p-value is

Using the Studentized residuals ti to determine influential points in a multiple regression model with k independent variable and n observations and letting tn-k-2 denote the upper critical value of a two-tail t test with a 0.10 level of significance, Xi is an influential point if

SCENARIO 13-10 You worked as an intern at We Always Win Car Insurance Company last summer.You notice that individual car insurance premiums depend very much on the age of the individual and the number of traffic tickets received by the individual.You performed a regression analysis in EXCEL and obtained the following partial information: Regression Statistics Multiple R 0.8546 R Square 0.7303 Adjusted R Square 0.6853 Standard Error 226.7502 Observations 15 ANOVA df SS MS F Significance F Regression 2 835284.6500 16.2457 0.0004 Residual 12 616987.8200 Total 2287557.1200 Coefficients Standard Error t Stat P-value Lower 99\% Upper 99\% Intercept 821.2617 161.9391 5.0714 0.0003 326.6124 1315.9111 Age -1.4061 2.5988 -0.5411 0.5984 -9.3444 6.5321 Tickets 243.4401 43.2470 5.6291 0.0001 111.3406 375.5396 -Referring to SCENARIO 13-10, the total degrees of freedom that are missing in the ANOVAtable should be .

SCENARIO 13-8 A financial analyst wanted to examine the relationship between salary (in $1,000) and 2 variables: age (X1 = Age) and experience in the field (X2 = Exper).He took a sample of 20 employees and obtained the following Microsoft Excel output: Regression Statistics Multiple R 0.8535 R Square 0.7284 Adjusted R Square 0.6964 Standard Error 10.5630 Observations 20 ANOYA df SS MS F Siqnificonce F Regression 2 5086.5764 2543.2882 22.7941 0.0000 Residual 17 1896.8050 111.5768 Total 19 6983.3814 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 1.5740 9.2723 0.1698 0.8672 -17.9888 21.1368 Age 1.3045 0.1956 6.6678 0.0000 0.8917 1.7173 Exper -0.1478 0.1944 -0.7604 0.4574 -0.5580 0.2624 -Referring to SCENARIO 13-8, the critical value of an F test on the entire regression for a level of significance of 0.01 is .

SCENARIO 13-18 A logistic regression model was estimated in order to predict the probability that a randomly chosen university or college would be a private university using information on mean total Scholastic Aptitude Test score (SAT) at the university or college and whether the TOEFL criterion is at least 90 (Toefl90 = 1 if yes, 0 otherwise.) The dependent variable, Y, is school type (Type = 1 if private and 0 otherwise).There are 80 universities in the sample. The PHStat output is given below: Binary Logistic Regression Predictor Coefficients SE Coef Z p -Value Intercept -3.9594 1.6741 -2.3650 0.0180 SAT 0.0028 0.0011 2.5459 0.0109 Toefl90:1 0.1928 0.5827 0.3309 0.7407 Deviance 101.9826 -Referring to SCENARIO 13-18, what is the p-value of the test statistic when testing whetherToefl90 makes a significant contribution to the model in the presence of SAT?

SCENARIO 13-19 The marketing manager for a nationally franchised lawn service company would like to study the characteristics that differentiate home owners who do and do not have a lawn service.A random sample of 30 home owners located in a suburban area near a large city was selected; 11 did not have a lawn service (code 0) and 19 had a lawn service (code 1).Additional information available concerning these 30 home owners includes family income (Income, in thousands of dollars) and lawn size (Lawn Size, in thousands of square feet). The PHStat output is given below: Binary Logistic Regression Predictor Coefficients SE Coef Z p -Value Intercept -7.8562 3.8224 -2.0553 0.0398 Income 0.0304 0.0133 2.2897 0.0220 Lawn Size 1.2804 0.6971 1.8368 0.0662 Deviance 25.3089 -Referring to SCENARIO 13-19, what is the estimated odds ratio for a home owner with a family income of $100,000 and a lawn size of 2,000 square feet?