Exam 15: Multiple Regression

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, the model that includes $X _ { 1 } , X _ { 2 } , X _ { 3 } , X _ { 5 } \text { and } X _ { 6 }$ should be selected using the adjusted $r ^ { 2 }$ statistic.

Collinearity is present if the dependent variable is linearly related to one of the explanatory variables.

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, the variable X1 should be dropped to remove collinearity?

The logarithm transformation can be used

SCENARIO 15-7-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 Scenario15-7-DataB.XLSX. 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 15-7-B, there is insufficient evidence to conclude that the quadratic term for the number of laborers is statistically significant at the 10% level of significance after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers.

SCENARIO 15-7-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 Scenario15-7-DataA.XLSX. 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 15-7-A, the model that includes only X2 and X3 should be selected using the adjusted $r ^ { 2 }$ statistic.

SCENARIO 15-4 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), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. -Referring to Scenario 15-4, what is the value of the test statistic to determine whether the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is significant at a 5% level of significance?

SCENARIO 15-7-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 Scenario15-7-DataA.XLSX. 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 15-7-A, there is reason to suspect collinearity between some pairs of predictors based on the values of the variance inflationary factor.

SCENARIO 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a curvilinear model to this data. The results obtained by Microsoft Excel follow -Referring to Scenario 15-3, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. The value of the test statistic is ________.

SCENARIO 15-7-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 Scenario15-7-DataB.XLSX. 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 15-7-B, the variable X3 should be dropped to remove collinearity?

SCENARIO 15-5 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 171 different vehicle models were collected: Accel Time: Acceleration time in sec. Cargo Vol: Cargo volume in cu. ft. HP: Horsepower MPG: Miles per gallon SUV: 1 if the vehicle model is an SUV with Coupe as the base when SUV and Sedan are both 0 Sedan: 1 if the vehicle model is a sedan with Coupe as the base when SUV and Sedan are both 0 The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 5 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.7461,0.5676,0.6764,0.8582,0.6632$ . -Referring to Scenario 15-5, what is the value of the variance inflationary factor of HP?

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, the variable X4 should be dropped to remove collinearity?

SCENARIO 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a curvilinear model to this data. The results obtained by Microsoft Excel follow -Referring to Scenario 15-3, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. The p-value of the test is ________.

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, the variable X2 should be dropped to remove collinearity?

SCENARIO 15-5 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 171 different vehicle models were collected: Accel Time: Acceleration time in sec. Cargo Vol: Cargo volume in cu. ft. HP: Horsepower MPG: Miles per gallon SUV: 1 if the vehicle model is an SUV with Coupe as the base when SUV and Sedan are both 0 Sedan: 1 if the vehicle model is a sedan with Coupe as the base when SUV and Sedan are both 0 The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 5 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.7461,0.5676,0.6764,0.8582,0.6632$ . -Referring to Scenario 15-5, what is the value of the variance inflationary factor of Cargo Vol?

SCENARIO 15-7-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 Scenario15-7-DataA.XLSX. 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 15-7-A, the variable $X _ { 3 }$ should be dropped to remove collinearity?

SCENARIO 15-3 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a curvilinear model to this data. The results obtained by Microsoft Excel follow -Referring to Scenario 15-3, suppose the chemist decides to use a t test to determine if the linear term is significant. The value of the test statistic is ______.

SCENARIO 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff $( Y )$ and the independent variables are the age of the worker $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ as the dependent variable and all other $X$ variables as independent variables are, respectively, $0.2628,0.1240,0.2404,0.3510,0.3342$ and $0.0993$ . The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Scenario 15-6, what is the value of the variance inflationary factor of Edu?

Collinearity will result in excessively low standard errors of the parameter estimates reported in the regression output.

SCENARIO 15-7-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 Scenario15-7-DataB.XLSX. 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 15-7-B, what is your decision on testing whether the quadratic term for land size is statistically significant at the 10% level of significance after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers?