Exam 14: Introduction to Multiple

14-30 Introduction to Multiple Regression -Referring to Scenario 14-7, the department head wants to use a t test to test for the significance of the coefficient of $X _ { 1 }$ . The value of the test statistic is ________.

SCENARIO 14-19 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 14-19, there is not enough evidence to conclude that the model is not a good-fitting model at a 0.05 level of significance.

14-30 Introduction to Multiple Regression -Referring to Scenario 14-7, the department head wants to use a t test to test for the significance of the coefficient of $X _ { 1 }$ . The p-value of the test is ________.

SCENARIO 14-16 What are the factors that determine the acceleration time (in sec.) from 0 to 60 miles per hour of a car? Data on the following variables for 30 different vehicle models were collected: $Y$ (Accel Time): Acceleration time in sec. $X _ { I }$ (Engine Size): c.c. $X _ { 2 }$ (Sedan): 1 if the vehicle model is a sedan and 0 otherwise The regression results using acceleration time as the dependent variable and the remaining variables as the independent variables are presented below. Regression Statistics Multiple R 0.6096 R Square 0.3716 Adjusted R Square 0.3251 Standard Error 1.4629 Observations 30 ANOVA Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept 7.1052 0.6574 10.8086 0.0000 5.7564 8.4540 Engine Size -0.0005 0.0001 -3.6477 0.0011 -0.0008 -0.0002 Sedan 0.7264 0.5564 1.3056 0.2027 -0.4152 1.8681 -Referring to Scenario 14-16, what is the p-value of the test statistic to determine whether being a sedan or not makes a significant contribution to the regression model in the presence of the other independent variable at a 5% level of significance?

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 is the p-value of the test statistic when testing whether mean teacher salary has any effect on percentage of students passing the proficiency test, taking into account the effect of instructional spending per pupil?

The slopes in a multiple regression model are called net regression coefficients.

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, what is the p-value for Wages?

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, there is sufficient evidence that instructional spending per pupil has an effect on percentage of students passing the proficiency test while holding constant the effect of mean teacher salary at a 5% level of significance.

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 aggregate price index has a positive impact on consumption, the p-value is

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.

SCENARIO 14-20-A 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-DataA.XLSX. S 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-A, what is the value of the test statistic when testing whether the number of milking cows has any effect on the total milk production while holding constant the effect of the other independent variable?

SCENARIO 14-4 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: -Referring to Scenario 14-4, ____% of the variation in the house size can be explained by the variation in the family income while holding the family size constant.

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, 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, taking into account the effect of mean teacher salary? a) $H _ { 0 } : \beta _ { 0 } = 0$ b) $H _ { 0 } : \beta _ { 1 } = 0$ c) $H _ { 0 } : \beta _ { 2 } = 0$ d) $H _ { 0 } : \beta _ { 3 } = 0$

14-30 Introduction to Multiple Regression -Referring to Scenario 14-7, the value of the adjusted coefficient of multiple determination, $r _ { \text {adj } } ^ { 2 }$ , 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, an employee who took 12 economics courses scores 10 on the performance rating. What is her estimated expected wage rate?

SCENARIO 14-4 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: -Referring to Scenario 14-4, what annual income (in thousands of dollars) would an individual with a family size of 4 need to attain a predicted 10,000 square foot home (House = 100)?

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 predicted consumption level for an economy with GDP equal to $4 billion and an aggregate price index of 150?

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, what are the predicted sales (in millions of dollars) for a company spending $100 million on capital and $100 million on wages?

SCENARIO 14-4 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: -Referring to Scenario 14-4, the partial F test for H0 : Variable X1 does not significantly improve the model after variable X2 has been included H1 : Variable X1 significantly improves the model after variable X2 has been included has ____ and ____ degrees of freedom.

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