Exam 13: Introduction to Multiple Regression

Instruction 13.8 A weight-loss clinic wants to use regression analysis to build a model for weight-loss of a client (measured in kilograms). Two variables thought to effect weight-loss are client's length of time on the weight loss program and time of session. These variables are described below: Y = Weight-loss (in kilograms) X₁ = Length of time in weight-loss program (in months) X₂ = 1 if morning session, 0 if not X₃ = 1 if afternoon session, 0 if not (Base level = evening session) Data for 12 clients on a weight-loss program at the clinic were collected and used to fit the interaction model: Y = ?0 + ?₁X₁ + ?₂X₂ + ?₃X₃ + ?₄X₁X₂ + ?₅X₁X₃ + ? Partial output from Microsoft Excel follows: Regression Statistics MultipleR 0.73514 R Square 0.540438 Adjusted R Square 0.157469 Standard Error 12.4147 Observations 12 ANOVA F=5.41118 Significance F= 0.040201 Coeff StdError t Stat p-value Intercept 0.089744 14.127 0.0060 0.9951 Length 6.22538 2.43473 2.54956 0.0479 Morn Ses 2.217272 22.1416 0.100141 0.9235 Aft Ses 11.8233 3.1545 3.558901 0.0165 Length* Morn Ses 0.77058 3.562 0.216334 0.8359 Length*Aft Ses -0.54147 3.35988 -0.161158 0.8773 -Referring to Instruction 13.8,what is the experimental unit for this analysis?

Instruction 13.16 A real estate builder wishes to determine how house size (House) is influenced by family income (Income), family size (Size) and education of the head of household (School). House size is measured in hundreds of square metres, income is measured in thousands of dollars and education is in years. The builder randomly selected 50 families and ran the multiple regression. Microsoft Excel output is provided below: OUTPUT SUMMARY Regression Statistics Multiple R 0.865 R Square 0.748 Adj. R Square 0.726 Std. Error 5.195 Observations 50 ANOVA df SS MS F Signiff Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff StdError t Stat p value Intercept -1.6335 5.8078 -0.281 0.7798 Income 0.4485 0.1137 3.9545 0.0003 Size 4.2615 0.8062 5.286 0.0001 School -0.6517 0.4319 -1.509 0.1383 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.16,which of the following values for the level of significance is the smallest for which at least two explanatory variables are significant individually?

Instruction 13.25 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), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married: 1 = married, 0 = otherwise), a dummy variable for head of household (Head: 1 = yes, 0 = no) and a dummy variable for management position (Manager: 1 = yes, 0 = no). We shall call this Model 1. Model 1 Regression Statistics Multiple R 0.7035 R Square 0.4949 Adj. R Square 0.4030 Std. Error 18.4861 Observations 40 ANOVA df SS MS F Signiff Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 223325.9 Coeff StdError tStat p value Lower 95\% Upper95\% Intercept 32.6595 23.18302 1.4088 0.1683 -14.5067 79.8257 Age 1.2915 0.3599 3.5883 0.0011 0.5592 2.0238 Edu -1.3537 1.1766 -1.1504 0.2582 -3.7476 1.0402 Job Yr 0.6171 0.5940 1.0389 0.3064 -0.5914 1.8257 Married -5.2189 7.6068 -0.6861 0.4974 -20.6950 10.2571 Head -14.2978 7.6479 -1.8695 0.0704 -29.8575 1.2618 Manager -24.8203 11.6932 -2.1226 0.0414 -48.6102 -1.0303 Model 2 is the regression analysis where the dependent variable is Unemploy and the independent variables are Age and Manager. The results of the regression analysis are given below: Mode 2 Regression Statistics Multiple R 0.6391 R Square 0.4085 Adj. R Square 0.3765 Std. Error 18.8929 Observations 40 ANOVA df SS MS F Signiff Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 Coeff StdError 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 Instruction 13.25 Model 1,you can conclude that,holding constant the effect of the other independent variables,the number of years of education received has no impact on the mean number of weeks a worker is unemployed due to a layoff at a 1% level of significance if all you have is the information of the 95% confidence interval estimate for ?₂.

A multiple regression is called 'multiple' because it has several data points.

Variance inflationary factor (VIF)measures the impact of collinearity among the Xs in a regression model.

When an additional explanatory variable is introduced into a multiple regression model,the adjusted r² can never decrease.

An interaction term in a multiple regression model may be used when the relationship between X₁ and Y changes for differing values of X₂.

The variation attribuInstruction to factors other than the relationship between the independent variables and the explained variable in a regression analysis is represented by

A regression had the following results: SST = 102.55,SSE = 82.04.It can be said that 20.0% of the variation in the dependent variable is explained by the independent variables in the regression.

The coefficient of multiple determination measures the proportion of the total variation in the dependent variable that is explained by a set of independent variables.

Instruction 13.35 The education department's regional executive officer wanted to predict the percentage of students passing a Grade 6 proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries) and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: -Referring to Instruction 13.35,which of the following is the correct null hypothesis to test whether daily mean of the percentage of students attending class has any effect on percentage of students passing the proficiency test,taking into account the effect of all the other independent variables?

Instruction 13.22 The education department's regional executive officer wanted to predict the percentage of students passing a Grade 6 proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries) and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: -Referring to Instruction 13.22,you can conclude that mean teacher salary has no impact on mean percentage of students passing the proficiency test at a 10% level of significance based solely on the 95% confidence interval estimate for β₂.

Instruction 13.31 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. Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.31,at the 0.01 level of significance,what conclusion should the microeconomist draw regarding the inclusion of Capital in the regression model?

Instruction 13.2 A lecturer in industrial relations believes that an individual's wage rate at a factory (Y) depends on his performance rating (X₁) and the number of economics courses the employee successfully completed at university (X₂). The lecturer randomly selects six workers and collects the following information: Employee Y(\ ) X1 X2 1 10 3 0 2 12 1 5 3 15 8 1 4 17 5 8 5 20 7 12 6 25 10 9 -Referring to Instruction 13.2,for these data,what is the value for the regression constant,b₀?

Instruction 13.19 You decide to predict petrol prices in different cities and towns in Australia for your term project. Your dependent variable is price of petrol per litre and your explanatory variables are per capita income, the number of firms that manufacture automobile parts in and around the city, the number of new business starts in the last year, population density of the city, percentage of local taxes on petrol and the number of people using public transportation. 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 Instruction 13.19,if variables that measure the number of new business starts in the last year and population density of the city were removed from the multiple regression model,which of the following would be true?

Instruction 13.35 The education department's regional executive officer wanted to predict the percentage of students passing a Grade 6 proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries) and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: -Referring to Instruction 13.35,the null hypothesis should be rejected at a 5% level of significance when testing whether daily mean of the percentage of students attending class has any effect on percentage of students passing the proficiency test,taking into account the effect of all the other independent variables.

In a multiple regression model,which of the following is correct regarding the value of the adjusted r²?

AU: Question 37 is the same as Question 36. Please check. Instruction 13.12 AU: Please advise if Instruction 13.12 can be renumbered to Instruction 13.11 and further questions renumbered. Or advise whether there shall be new Instruction 13.11 included. The Head of the Accounting Department wanted to see if she could predict the average grade of students using the number of course units (credits) and total university entrance exam scores of each. She takes a sample of students and generates the following Microsoft Excel output: OUTPUT SUMMARY Regression Statistics MultipleR 0.916 R Square 0.839 Adj. R Square 0.732 Std. Error 0.24685 Observations 6 ANOVA df SS MS F Signiff 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 GDP -0.247270 0.06268485 -3.945 0.0290 Price 0.001443 0.00101241 1.425 0.2494 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 13.12,the predicted mean grade for a student carrying 15 course units and who has a total university entrance exam score of 1,100 is ___________.

Instruction 13.6 One of the most common questions of prospective house buyers pertains to the average cost of heating in dollars (Y). To provide its customers with information on that matter, a large real estate firm used the following four variables to predict heating costs: the daily minimum outside temperature in degrees of Celsius (X₁), the amount of insulation in cm (X₂), the number of windows in the house (X₃) and the age of the furnace in years (X₄). Given below are the Microsoft Excel outputs of two regression models. -Referring to Instruction 13.6,the estimated value of the partial regression parameter β₁ in Model 1 means that

Instruction 13.22 The education department's regional executive officer wanted to predict the percentage of students passing a Grade 6 proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily average of the percentage of students attending class (% Attendance), average teacher salary in dollars (Salaries) and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Following is the multiple regression output with Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending: -Referring to Instruction 13.22,the alternative hypothesis H₁: At least one of β_j ≠ 0 for j = 1,2,3 implies that percentage of students passing the proficiency test is affected by at least one of the explanatory variables.