Exam 14: Introduction to Multiple

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, what is the 95% confidence interval for the expected change in heating costs as a result of a 1 degree Fahrenheit change in the daily minimum outside Temperature?

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 that instructional spending per pupil has no impact on the mean percentage of students passing the proficiency test, taking into account the effect of mean teacher salary, at a 5% level of significance using the confidence interval estimate for $\beta _ { 2 }$ .

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, you can conclude that, holding constant the effect of the other independent variable, age has no impact on the mean number of weeks a worker is unemployed due to a layoff at a 5% level of significance if we use only the information 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.

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, we can conclude that, holding constant the effect of the other independent variable, there is a difference in the mean number of weeks a worker is unemployed due to a layoff between a worker who is in a management position and one who is not at a 5% level of significance if we use only the information of the 95% confidence interval estimate for the difference in the mean number of weeks a worker is unemployed due to a layoff between a worker who is in a management position and one who is not.

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 $X _ { 1 }$ . For a level of significance of 0.05, the critical values of the test are ________.

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 null hypothesis should be rejected at a 5% level of significance when testing 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.

The coefficient of multiple determination $r _ { Y 1 \cdot 2 } ^ { 2 }$ measures the proportion of variation in Y that is explained by $X _ { 1 } \text { and } X _ { 2 }$ .

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, when the economist used a simple linear regression model with consumption as the dependent variable and GDP as the independent variable, he obtained an $r ^ { 2 }$ value of 0.971. What additional percentage of the total variation of consumption has been Explained by including aggregate prices in the multiple regression?

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, which of the following is a correct interpretation for the coefficient of partial determination $r _ { Y 1.23 } ^ { 2 }$ ?

If you have taken into account all relevant explanatory factors, the residuals from a multiple regression model should be random.

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 estimated change in the mean salary (in $1,000) for an employee who has one additional year of experience holding age constant is ________.

SCENARIO 14-12 As a project for his business statistics class, a student examined the factors that determined parking meter rates throughout the campus area. Data were collected for the price (\$) per hour of parking, blocks to the quadrangle, and whether the parking is on or off campus. The population regression model hypothesized is $Y _ { i } = \alpha + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + \varepsilon$ where $Y$ is the meter price per hour $X _ { l }$ is the number of blocks to the quad $X _ { 2 }$ is a dummy variable that takes the value 1 if the meter is located on campus and 0 otherwise The following Excel results are obtained. Regression Statistics Multiple R 0.5536 R Square 0.3064 Adjusted R Square 0.2812 Standard Error 0.4492 Observations 58 ANOVA df SS MS F Significance F Regression 2 4.9035 2.4518 12.1501 0.0000 Residual 55 11.0984 0.2018 Total 57 16.0019 Coefficients Standard Error Stat P-value Lower 99\% Upper 99\% Intercept 1.6500 0.2028 8.1359 0.0000 1.1089 2.1912 Block -0.2504 0.0529 -4.7355 0.0000 -0.3915 -0.1093 Campus 0.1552 0.1297 1.1966 0.2366 -0.1908 0.5011 -Referring to Scenario 14-12, what is the correct interpretation for the estimated coefficient for $X _ { 2 }$ ?

SCENARIO 14-10 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 $\text { ANOVA }$ Coefficients Standard Error tStat 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 14-10, to test the significance of the multiple regression model, the p- value of the test statistic in the sample is ______.

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, which of the following is the correct alternative hypothesis to test whether the number of milking cows has any effect on the total milk production while Holding constant the effect of the other independent variables? 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 14-4 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, which of the following values for the level of significance is the smallest for which at least one explanatory variable is significant individually?

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 $H _ { 0 }$ : Variable $X _ { 2 }$ does not significantly improve the model after variable $X _ { 1 }$ has been included $H _ { 1 }$ : Variable $X _ { 2 }$ significantly improves the model after variable $X _ { I }$ has been included has and degrees of freedom. .

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 null hypothesis should be rejected at a 5% level of significance when testing whether there is a significant relationship between percentage of students passing the proficiency test and the entire set of explanatory 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, construct the residual plot for the number of laborers.

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 all of the explanatory variables affect 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 14-3 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 for the significance of the coefficient on aggregate price index, the p-value is