Exam 14: Introduction to Multiple

SCENARIO 14-20-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-B, which of the following is the correct alternative hypothesis to test whether the number of laborers has any effect on the total milk production while holding Constant the effect of the other independent variables? S a) $H _ { 1 } : \beta _ { 1 } \neq 0$ b) $H _ { 1 } : \beta _ { 2 } \neq 0$ c) $H _ { 1 } : \beta _ { 3 } \neq 0$ d) $H _ { 1 } : \beta _ { 1 } \neq \beta _ { 2 } \neq \beta _ { 3 } \neq 0$

SCENARIO 14-13 An econometrician is interested in evaluating the relationship of demand for building materials to mortgage rates in Los Angeles and San Francisco. He believes that the appropriate model is $Y = 10 + 5 X _ { 1 } + 8 X _ { 2 }$ where where = mortgage rate in \% =1 if SF, 0 if LA Y= demand in \ 100 per capita -Referring to Scenario 14-13, the effect of living in San Francisco rather than Los Angeles is to increase the mean demand by an estimated ________.

SCENARIO 14-20-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-B, the lower and upper limits of the 95% prediction interval for the total milk production of a farm with 40 milking cows, 30 acres of land and 3 laborers are _____ liters and _____ liters, respectively.

SCENARIO 14-20-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-B, construct the normal probability plot for the regression residuals.

14-22 Introduction to Multiple Regression One of the most common questions of prospective house buyers pertains to the cost of heating in dollars $( Y )$ . To provide its customers with information on that matter, a large real estate firm used the following 2 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit $\left( X _ { 1 } \right)$ and the amount of insulation in inches $\left( X _ { 2 } \right)$ . Given below is EXCEL output of the regression model. Regression Statistics Multiple R 0.5270 R Square 0.2778 Adjusted R Square 0.1928 Standard Error 40.9107 Observations 20 ANOVA Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 448.2925 90.7853 4.9379 0.0001 256.7522 639.8328 Temperature -2.7621 1.2371 -2.2327 0.0393 -5.3721 -0.1520 Insulation -15.9408 10.0638 -1.5840 0.1316 -37.1736 Also $\operatorname { SSR } \left( X _ { 1 } \mid X _ { 2 } \right) = 8343.3572$ and $\operatorname { SSR } \left( X _ { 2 } \mid X _ { 1 } \right) = 4199.2672$ -Referring to Scenario 14-6, the value of the partial F test statistic is ____ for $H _ { 0 }$ : Variable $X _ { 1 }$ does not significantly improve the model after variable $X _ { 2 }$ has been included $H _ { 1 }$ : Variable $X _ { 1 }$ significantly improves the model after variable $X _ { 2 }$ has been included

SCENARIO 14-3 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 14-3, what is the estimated mean consumption level for an economy with GDP equal to $4 billion and an aggregate price index of 150?

SCENARIO 14-20-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-B, there is sufficient evidence that at least one of the explanatory variables is related to total milk production at a 1% level of significance when testing whether there is a significant relationship between total milk production and the entire set of explanatory variables.

SCENARIO 14-8 A financial analyst wanted to examine the relationship between salary (in $\$ 1,000$ ) and 2 variables: age $\left(X_{1}=\mathrm{Age}\right)$ and experience in the field $\left(X_{2}=\right.$ 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 $\text { ANOVA }$ Coefficients Standard Error t Stat P-value Lower 95\% O5\% 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 Also the sum of squares due to the regression for the model that includes only Age is 5022.0654 while the sum of squares due to the regression for the model that includes only Exper is 125.9848. -Referring to Scenario 14-8, the value of the partial F test statistic is ____ for $H _ { 0 }$ : Variable $X _ { 1 }$ does not significantly improve the model after variable $X _ { 2 }$ has been included $H _ { 1 }$ : Variable $X _ { 1 }$ significantly improves the model after variable $X _ { 2 }$ has been included

SCENARIO 14-20-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-B, the lower and upper limits of the 95% confidence interval for the mean total milk production of a farm with 30 milking cows, 20 acres of land and 3 laborers are _____ liters and _____ liters, respectively.

SCENARIO 14-5 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. -Referring to Scenario 14-5, 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?

SCENARIO 14-3 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 14-3, to test whether gross domestic product has a positive impact on consumption, the p-value is

SCENARIO 14-2 A professor of industrial relations believes that an individual's wage rate at a factory $( Y )$ depends on his performance rating $\left( X _ { 1 } \right)$ and the number of economics courses the employee successfully completed in college $\left( X _ { 2 } \right)$ . The professor randomly selects 6 workers and collects the following information: -Referring to Scenario 14-2, for these data, what is the value for the regression constant, b0?

SCENARIO 14-17 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 $\text { ANOVA }$ 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 14-17, what is the 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 14-15 The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), mean teacher salary in thousands of dollars (Salaries), and instructional spending per pupil in thousands of dollars (Spending) of 47 schools in the state. Following is the multiple regression output with $Y = \%$ Passing as the dependent variable, $X _ { 1 } =$ Salaries and $X _ { 2 } =$ Spending: Regression Statistics Multiple R 0.4276 R Square 0.1828 Adjusted R Square 0.1457 Standard Error 5.7351 Observations 47 ANOVA Coefficients Standard Error t Stat \rho -value Lower 95\% Upper 95\% Intercept -72.9916 45.9106 -1.5899 0.1190 -165.5184 19.5352 Salary 2.7939 0.8974 3.1133 0.0032 0.9853 4.6025 Spending 0.3742 0.9782 0.3825 0.7039 -1.5972 2.3455 -Referring to Scenario 14-15, what are the lower and upper limits of the 95% confidence interval estimate for the effect of a one thousand dollar increase in mean teacher salary on the mean percentage of students passing the proficiency test?

SCENARIO 14-18 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). 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 14-18, the null hypothesis that the model is a good-fitting model cannot be rejected when allowing for a 5% probability of making a type I error.

SCENARIO 14-15 The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), mean teacher salary in thousands of dollars (Salaries), and instructional spending per pupil in thousands of dollars (Spending) of 47 schools in the state. Following is the multiple regression output with $Y = \%$ Passing as the dependent variable, $X _ { 1 } =$ Salaries and $X _ { 2 } =$ Spending: Regression Statistics Multiple R 0.4276 R Square 0.1828 Adjusted R Square 0.1457 Standard Error 5.7351 Observations 47 ANOVA Coefficients Standard Error t Stat \rho -value Lower 95\% Upper 95\% Intercept -72.9916 45.9106 -1.5899 0.1190 -165.5184 19.5352 Salary 2.7939 0.8974 3.1133 0.0032 0.9853 4.6025 Spending 0.3742 0.9782 0.3825 0.7039 -1.5972 2.3455 -Referring to Scenario 14-15, you can conclude definitively that mean teacher salary individually has no impact on the mean percentage of students passing the proficiency test, taking into account the effect of that instructional spending per pupil, at a 10% level of significance based solely on but not actually computing the 90% confidence interval estimate for $\beta _ { 1 }$ .

SCENARIO 14-15 The superintendent of a school district wanted to predict the percentage of students passing a sixth- grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), mean teacher salary in thousands of dollars (Salaries), and instructional spending per pupil in thousands of dollars (Spending) of 47 schools in the state. Following is the multiple regression output with $Y = \%$ Passing as the dependent variable, $X _ { 1 } =$ Salaries and $X _ { 2 } =$ Spending: Regression Statistics Multiple R 0.4276 R Square 0.1828 Adjusted R Square 0.1457 Standard Error 5.7351 Observations 47 ANOVA Coefficients Standard Error t Stat \rho -value Lower 95\% Upper 95\% Intercept -72.9916 45.9106 -1.5899 0.1190 -165.5184 19.5352 Salary 2.7939 0.8974 3.1133 0.0032 0.9853 4.6025 Spending 0.3742 0.9782 0.3825 0.7039 -1.5972 2.3455 -Referring to Scenario 14-15, the alternative hypothesis $H _ { 1 } : \text { At least one of } \beta _ { j } \neq 0 \text { for } j = 1,2$ implies that percentage of students passing the proficiency test is related to at least one of the explanatory variables.

SCENARIO 14-17 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 $\text { ANOVA }$ 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 14-17, which of the following is a correct statement?

SCENARIO 14-17 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 $\text { ANOVA }$ 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 14-17, what are the lower and upper limits of the 95% confidence interval estimate for the effect of a one year increase in age on the mean number of weeks a worker is unemployed due to a layoff after taking into consideration the effect of all the other independent variables?

SCENARIO 14-20-B You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario14-20-DataB.XLSX. MILK 84686 101876 103248 70508 76072 86615 87508 105195 120351 68658 You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 14-20-B, what is the value of the coefficient of partial determination $r _ { Y 3.12 } ^ { 2 }$ ?