Instruction 13-9 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) XS1U1B11 S1U1B0= Length of time in weight-loss program (in months) XS1U1B12S1U1B0 = 1 if morning session,0 if not XS1U1B13S1U1B0 = 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 = β0 + βS1U1B11S1U1B0XS1U1B11S1U1B0 + βS1U1B12S1U1B0XS1U1B12S1U1B0 + βS1U1B13S1U1B0XS1U1B13S1U1B0 + βS1U1B14S1U1B0XS1U1B11S1U1B0XS1U1B12S1U1B0 + βS1U1B15S1U1B0XS1U1B11S1U1B0XS1U1B13S1U1B0 + ε Partial output from Microsoft Excel follows: Regression Statistics $\begin{array} { l l } \text { Multiple R } & 0.73514 \\ \text { R Square } & 0.540438 \\ \text { Adjusted R Square } & 0.157469 \\ \text { Standard Error } & 12.4147 \\ \text { Observations } & 12 \end{array}$ ANOVA $\begin{array} { c c c c c } F = 5.41118 & \text { Significance } F = 0.040201 & & \\ & & & & \\ \text { Intercept } & \text { Coeff } & \text { StdError } & t \text { Stat } & P \text {-value } \\ \text { Length } \left( X _ { 1 } \right) & 0.089744 & 14.127 & 0.0060 & 0.9951 \\ \text { Morn Ses } \left( X _ { 2 } \right) & 6.22538 & 2.43473 & 2.54956 & 0.0479 \\ \text { Aft Ses } \left( X _ { 3 } \right) & 2.217272 & 22.1416 & 0.100141 & 0.9235 \\ \text { Length*Morn Ses } & 11.8233 & 3.1545 & 3.558901 & 0.0165 \\ \text { Length*Aft Ses } & 0.77058 & 3.562 & 0.216334 & 0.8359 \\ & - 0.54147 & 3.35988 & - 0.161158 & 0.8773 \end{array}$ -Referring to Instruction 13-9,in terms of the ?'s in the model,give the mean change in weight-loss (Y)for every 1 month increase in time in the program (X1)when attending the evening session.

A) ?1 + ?4 B) ?1 + ?5 C) ?1 D) ?4 + ?5 A) ?1 + ?4 B) ?1 + ?5 C) ?1 D) ?4 + ?5

Instruction 13-13 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, XS1U1B11S1U1B0 = % Attendance,XS1U1B12 S1U1B0= Salaries and XS1U1B13 S1U1B0= Spending: $\begin{array}{lr} \hline {\text { Regression Statistics }} \\ \hline \text { Multiple R } & 0.7930 \\ \text { R Square } & 0.6288 \\ \text { Adjusted R } & 0.6029 \\ \text { Square } & \\ \text { Standard } & 10.4570 \\ \text { Error } & \\ \text { Observations } & 47 \\ \hline \end{array}$ ANOVA $\begin{array}{lrrrrr} \hline & d f & {\text { SS }} & M S & F & \text { Significance } F \\ \text { Regression } & 3 & 7965.08 & 2655.03 & 24.2802 & 0.0000 \\ \text { Residual } & 43 & 4702.02 & 109.35 & & \\ \text { Total } & 46 & 12667.11 & & & \end{array}$ $\begin{array}{lrrrrrrr} \hline & \text { Coefficients } & \begin{array}{c} \text { Standard } \\ \text { Error } \end{array} & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \\ \hline \text { Intercept } & -753.4225 & 101.1149 & -7.4511 & 0.0000 & -957.3401 & -549.5050 \\ \text { \% Attendance } & 8.5014 & 1.0771 & 7.8929 & 0.0000 & 6.3292 & 10.6735 \\ \text { Salary } & 0.000000685 & 0.0006 & 0.0011 & 0.9991 & -0.0013 & 0.0013 \\ \text { Spending } & 0.0060 & 0.0046 & 1.2879 & 0.2047 & -0.0034 & 0.0153 \end{array}$ -Referring to Instruction 13-13,which of the following is the correct alternative 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?

A) H1: ?0 ? 0 B) H1: ?2 ? 0 C) H1: ?3 ? 0 D) H1: ?1 ? 0 A) H1: ?0 ? 0 B) H1: ?2 ? 0 C) H1: ?3 ? 0 D) H1: ?1 ? 0

Instruction 13-8 You worked as an intern at We Always Win Car Insurance Company last summer.You notice that individual car insurance premium depends very much on the age of the individual,the number of traffic tickets received by the individual,and the population density of the city in which the individual lives.You performed a regression analysis in Microsoft Excel and obtained the following information: $\begin{array}{l} \text { Regression Analysis }\\ \begin{array}{lr} \hline {\text { Regression Statistics }} \\ \hline \text { Multiple R } & 0.63 \\ \text { R Square } & 0.40 \\ \text { Adjusted R Square } & 0.23 \\ \text { Standard Error } & 50.00 \\ \text { Observations } & 15.00 \end{array} \end{array}$ ANOVA $\begin{array}{lrrrrrr} & d f & & S S & M S & F & \text { Significance } F \\ \hline \text { Regression } & 3 & & 5994.24 & 2.40 & 0.12 \\ \text { Residual } & 11 & 27496.82 & & & \\ \text { Total } & & 45479.54 & & & \end{array}$ $\begin{array}{lrrrrrrr} & \text { Coefficients } & \text { Standard } & t \text { Stat } & \text { P-value } & \text { Lower 99.0\% } & \text { Upper 99.0\% } \\ \hline \text { Intercept } & {\text { Error }} & & & & & \\ \text { AGE } & 123.80 & 48.71 & 2.54 & 0.03 & -27.47 & 275.07 \\ \text { TICKETS } & -0.82 & 0.87 & -0.95 & 0.36 & -3.51 & 1.87 \\ \text { DENSITY } & 21.25 & 10.66 & 1.99 & 0.07 & -11.86 & 54.37 \\ & -3.14 & 6.46 & -0.49 & 0.64 & -23.19 & 16.91 \end{array}$ -Referring to Instruction 13-8,the 99% confidence interval for the change in average insurance premiums of a person who has become one year older (i.e. ,the slope coefficient for AGE)is ________.

The answer of Instruction 13-8 You worked as an intern at...

Instruction 13-4 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 $ \begin{array}{ll}\text { Multiple R } & 0.865 \\ \text { R Square } & 0.748 \\ \text { Adj. R Square } & 0.726 \\ \text { Std. Error } & 5.195 \\ \text { Observations } & 50\end{array} $ ANOVA $\begin{array}{llllll} & d f & \text { SS } & \text { MS } & F & \text { Siguif } F \\ \text { Regression } & & 3605.7736 & 901.4434 & & 0.0001 \\ \text { Residual } & & 1214.2264 & 26.9828 & & \\ \text { Total } & 49 & 4820.0000 & & & \end{array}$ $\begin{array}{lllll} & \text { Coeff } & \text { StdError } & \boldsymbol{t} \text { Stat } & \boldsymbol{P} \text {-value } \\ \text { Intercept } & -1.6335 & 5.8078 & -0.281 & 0.7798 \\ \text { Income } & 0.4485 & 0.1137 & 3.9545 & 0.0003 \\ \text { Size } & 4.2615 & 0.8062 & 5.286 & 0.0001 \\ \text { School } & -0.6517 & 0.4319 & -1.509 & 0.1383 \end{array}$ Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 13-4,one individual in the sample had an annual income of $100,000,a family size of 10,and an education of 16 years.This individual owned a home with an area of 7,000 square metre (House = 70.00).What is the residual (in hundreds of square metre)for this data point?

A) 7.40 B) 2.52 C) -5.40 D) -2.52 A) 7.40 B) 2.52 C) -5.40 D) -2.52

Instruction 13-5 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. SUMMARY Regression Statistics \begin{tabular} { l r } Multiple $\mathrm { R }$ & $0.830$ \\ \hline Square & $0.689$ \end{tabular} R Square $\quad 0.689$ Adj. R Square $\quad 0.662$ Std. Error $\quad 17501.643$ Observations 26 ANOVA $\begin{array} { l l l l l l } & \boldsymbol { d f } & \text { SS } & \text { MS } & \boldsymbol { F } & \text { Siguif } \boldsymbol { F } \\ \text { Regression } & 2 & 15579777040 & 7789888520 & 25.432 & 0.0001 \\ \text { Residual } & 23 & 7045072780 & 306307512 & & \\ \text { Total } & 25 & 22624849820 & & & \end{array}$ $\begin{array} { l l l l l } & \text { Coeff } & \text { StdError } & \boldsymbol { t } \text { Stat } & \boldsymbol { P } \text {-value } \\ \text { Intercept } & 15800.0000 & 6038.2999 & 2.617 & 0.0154 \\ \text { Capital } & 0.1245 & 0.2045 & 0.609 & 0.5485 \\ \text { Wages } & 7.0762 & 1.4729 & 4.804 & 0.0001 \end{array}$ Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 13-5,which of the following values for ? is the smallest for which the regression model as a whole is significant?

A) 0.001 B) 0.01 C) 0.05 D) 0.00005 A) 0.001 B) 0.01 C) 0.05 D) 0.00005

Instruction 13-2 A lecturer in industrial relations believes that an individual's wage rate at a factory (Y)depends on his performance rating (XS1U1B11S1U1B0)and the number of economics courses the employee successfully completed at university (XS1U1B12S1U1B0).The lecturer randomly selects six workers and collects the following information: $\begin{array}{clll} \underline{\text { Employee }} & \underline{Y(\$)} & \underline{X}_{1} &\underline{X}_{2}\\ 1 & 10 & 3 & 0 \\ 2 & 12 & 1 & 5 \\ 3 & 15 & 8 & 1 \\ 4 & 17 & 5 & 8 \\ 5 & 20 & 7 & 12 \\ 6 & 25 & 10 & 9 \end{array}$ -Referring to Instruction 13-2,for these data,what is the value for the regression constant,b0?

A) 9.103 B) 6.932 C) 1.054 D) 0.616 A) 9.103 B) 6.932 C) 1.054 D) 0.616

Instruction 13-4 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 $ \begin{array}{ll}\text { Multiple R } & 0.865 \\ \text { R Square } & 0.748 \\ \text { Adj. R Square } & 0.726 \\ \text { Std. Error } & 5.195 \\ \text { Observations } & 50\end{array} $ ANOVA $\begin{array}{llllll} & d f & \text { SS } & \text { MS } & F & \text { Siguif } F \\ \text { Regression } & & 3605.7736 & 901.4434 & & 0.0001 \\ \text { Residual } & & 1214.2264 & 26.9828 & & \\ \text { Total } & 49 & 4820.0000 & & & \end{array}$ $\begin{array}{lllll} & \text { Coeff } & \text { StdError } & \boldsymbol{t} \text { Stat } & \boldsymbol{P} \text {-value } \\ \text { Intercept } & -1.6335 & 5.8078 & -0.281 & 0.7798 \\ \text { Income } & 0.4485 & 0.1137 & 3.9545 & 0.0003 \\ \text { Size } & 4.2615 & 0.8062 & 5.286 & 0.0001 \\ \text { School } & -0.6517 & 0.4319 & -1.509 & 0.1383 \end{array}$ Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 13-4,at the 0.01 level of significance,what conclusion should the builder draw regarding the inclusion of School in the regression model?

A) School is not significant in explaining house size and should not be included in the model because its p-value is more than 0.01. B) School is not significant in explaining house size and should not be included in the model because its p-value is less than 0.01. C) School is significant in explaining house size and should be included in the model because its p-value is more than 0.01. D) School is significant in explaining house size and should be included in the model because its p-value is less than 0.01. A) School is not significant in explaining house size and should not be included in the model because its p-value is more than 0.01. B) School is not significant in explaining house size and should not be included in the model because its p-value is less than 0.01. C) School is significant in explaining house size and should be included in the model because its p-value is more than 0.01. D) School is significant in explaining house size and should be included in the model because its p-value is less than 0.01.

Exam 13: Introduction to Multiple Regression

Instruction 13-13 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: Regression Statistics Multiple R 0.7930 R Square 0.6288 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 ANOVA df SS MS F Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 0.0000 -957.3401 -549.5050 \% Attendance 8.5014 1.0771 7.8929 0.0000 6.3292 10.6735 Salary 0.000000685 0.0006 0.0011 0.9991 -0.0013 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 -Referring to Instruction 13-13,what is the value of the test statistics when testing whether instructional spending per pupil has any effect on percentage of students passing the proficiency test,taking into account the effect of all the other independent variables?

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Question 41

Instruction 13-9 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 = β0 + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₁X₂ + β₅X₁X₃ + ε Partial output from Microsoft Excel follows: Regression Statistics Multiple R 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 Intercept Coeff StdError t Stat P -value Length 0.089744 14.127 0.0060 0.9951 Morn Ses 6.22538 2.43473 2.54956 0.0479 Aft Ses 2.217272 22.1416 0.100141 0.9235 LengthMorn Ses 11.8233 3.1545 3.558901 0.0165 LengthAft Ses 0.77058 3.562 0.216334 0.8359 -0.54147 3.35988 -0.161158 0.8773 -Referring to Instruction 13-9,the overall model for predicting weight-loss (Y)is statistically significant at the 0.05 level.

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Question 42

Instruction 13-9 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 = β0 + β₁X₁ + β₂X₂ + β₃X₃ + β₄X₁X₂ + β₅X₁X₃ + ε Partial output from Microsoft Excel follows: Regression Statistics Multiple R 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 Intercept Coeff StdError t Stat P -value Length 0.089744 14.127 0.0060 0.9951 Morn Ses 6.22538 2.43473 2.54956 0.0479 Aft Ses 2.217272 22.1416 0.100141 0.9235 LengthMorn Ses 11.8233 3.1545 3.558901 0.0165 LengthAft Ses 0.77058 3.562 0.216334 0.8359 -0.54147 3.35988 -0.161158 0.8773 -Referring to Instruction 13-9,in terms of the ?'s in the model,give the mean change in weight-loss (Y)for every 1 month increase in time in the program (X₁)when attending the evening session.

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Question 43

Instruction 13-13 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: Regression Statistics Multiple R 0.7930 R Square 0.6288 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 ANOVA df SS MS F Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 0.0000 -957.3401 -549.5050 \% Attendance 8.5014 1.0771 7.8929 0.0000 6.3292 10.6735 Salary 0.000000685 0.0006 0.0011 0.9991 -0.0013 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 -Referring to Instruction 13-13,which of the following is the correct alternative 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?

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Question 44

Instruction 13-8 You worked as an intern at We Always Win Car Insurance Company last summer.You notice that individual car insurance premium depends very much on the age of the individual,the number of traffic tickets received by the individual,and the population density of the city in which the individual lives.You performed a regression analysis in Microsoft Excel and obtained the following information: Regression Analysis Regression Statistics Multiple R 0.63 R Square 0.40 Adjusted R Square 0.23 Standard Error 50.00 Observations 15.00 ANOVA df SS MS F Significance F Regression 3 5994.24 2.40 0.12 Residual 11 27496.82 Total 45479.54 Coefficients Standard t Stat P-value Lower 99.0\% Upper 99.0\% Intercept Error AGE 123.80 48.71 2.54 0.03 -27.47 275.07 TICKETS -0.82 0.87 -0.95 0.36 -3.51 1.87 DENSITY 21.25 10.66 1.99 0.07 -11.86 54.37 -3.14 6.46 -0.49 0.64 -23.19 16.91 -Referring to Instruction 13-8,the 99% confidence interval for the change in average insurance premiums of a person who has become one year older (i.e. ,the slope coefficient for AGE)is ________.

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Question 45

Instruction 13-14 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: SUMMARY Regression Statistics Multiple R 0.916 R Square 0.839 Adj. R Square 0.732 Std. Error 0.24685 Observations 6 ANOVA SS MS Signif Regression 2 0.95219 0.47610 7.813 0.0646 Residual 3 0.18281 0.06094 Total 5 1.13500 Coeff StdError Stat -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 13-14,the Head of Department wants to test H₀: β₁ = β₂ = 0.The critical value of the F test for a level of significance of 0.05 is ________.

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Question 46

Instruction 13-16 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. Regression Statistics Multiple R 0.7035 R Square 0.4949 Adjusted R 0.4030 Square Standard 18.4861 Error Observations 40 ANOVA df SS MS F Significance F Regression 6 11048.6415 1841.4402 5.3885 0.00057 Residual 33 11277.2586 341.7351 Total 39 22325.9 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% 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: Regression Statistics Multiple R 0.6391 R Square 0.4085 Adjusted R 0.3765 Square Standard Error 18.8929 Observations 40 ANOVA df SS MS F Significance F Regression 2 9119.0897 4559.5448 12.7740 0.0000 Residual 37 13206.8103 356.9408 Total 39 22325.9 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 Instruction 13-16 Model 1,the null hypothesis H₀: ?₁ = ?₂ = ?₃ = ?₄ = ?₅ = ?₆ = 0 implies that the number of weeks a worker is unemployed due to a layoff is not related to one of the explanatory variables.

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Question 47

The total sum of squares (SST)in a regression model will never exceed the regression sum of squares (SSR).

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Question 48

Instruction 13-13 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: Regression Statistics Multiple R 0.7930 R Square 0.6288 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 ANOVA df SS MS F Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 0.0000 -957.3401 -549.5050 \% Attendance 8.5014 1.0771 7.8929 0.0000 6.3292 10.6735 Salary 0.000000685 0.0006 0.0011 0.9991 -0.0013 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 -Referring to Instruction 13-13,the alternative hypothesis H₁: At least one of ?_j ? 0 for j = 1,2,3 implies that percentage of students passing the proficiency test is affected by all of the explanatory variables.

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Question 49

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

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Question 50

Consider a regression in which b₂ = -1.5 and the standard error of this coefficient equals 0.3.To determine whether X₂ is a significant explanatory variable,you would compute an observed t-value of -5.0.

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Question 51

Instruction 13-13 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: Regression Statistics Multiple R 0.7930 R Square 0.6288 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 ANOVA df SS MS F Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 0.0000 -957.3401 -549.5050 \% Attendance 8.5014 1.0771 7.8929 0.0000 6.3292 10.6735 Salary 0.000000685 0.0006 0.0011 0.9991 -0.0013 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 -Referring to Instruction 13-13,what is the p-value of the test statistic to determine whether there is a significant relationship between percentage of students passing the proficiency test and the entire set of explanatory variables?

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Question 52

Instruction 13-13 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: Regression Statistics Multiple R 0.7930 R Square 0.6288 Adjusted R 0.6029 Square Standard 10.4570 Error Observations 47 ANOVA df SS MS F Significance F Regression 3 7965.08 2655.03 24.2802 0.0000 Residual 43 4702.02 109.35 Total 46 12667.11 Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 0.0000 -957.3401 -549.5050 \% Attendance 8.5014 1.0771 7.8929 0.0000 6.3292 10.6735 Salary 0.000000685 0.0006 0.0011 0.9991 -0.0013 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 -Referring to Instruction 13-13,what is the p-value of the test statistics 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?

(Short Answer)

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Question 53

Instruction 13-4 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 Siguif F Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff StdError Stat -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 13-4,one individual in the sample had an annual income of $100,000,a family size of 10,and an education of 16 years.This individual owned a home with an area of 7,000 square metre (House = 70.00).What is the residual (in hundreds of square metre)for this data point?

(Multiple Choice)

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Question 54

Instruction 13-5 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. SUMMARY Regression Statistics \begin{tabular} { l r } Multiple $\mathrm { R }$ & $0.830$ \\ \hline Square & $0.689$ \end{tabular} R Square $\quad 0.689$ Adj. R Square $\quad 0.662$ Std. Error $\quad 17501.643$ Observations 26 ANOVA SS MS Siguif Regression 2 15579777040 7789888520 25.432 0.0001 Residual 23 7045072780 306307512 Total 25 22624849820 Coeff StdError Stat -value Intercept 15800.0000 6038.2999 2.617 0.0154 Capital 0.1245 0.2045 0.609 0.5485 Wages 7.0762 1.4729 4.804 0.0001 Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 13-5,which of the following values for ? is the smallest for which the regression model as a whole is significant?

(Multiple Choice)

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Question 55

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: $\begin{array}{clll}\underline{\text { Employee }} & \underline{Y(\$)} & \underline{X}_{1} &\underline{X}_{2}\\1 & 10 & 3 & 0 \\2 & 12 & 1 & 5 \\3 & 15 & 8 & 1 \\4 & 17 & 5 & 8 \\5 & 20 & 7 & 12 \\6 & 25 & 10 & 9\end{array}$ -Referring to Instruction 13-2,for these data,what is the value for the regression constant,b₀?

(Multiple Choice)

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Question 56

When a dummy variable is included in a multiple regression model,the interpretation of the estimated slope coefficient does not make any sense anymore.

(True/False)

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Question 57

Instruction 13-4 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 Siguif F Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff StdError Stat -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 13-4,at the 0.01 level of significance,what conclusion should the builder draw regarding the inclusion of School in the regression model?

(Multiple Choice)

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Question 58

When an additional explanatory variable is introduced into a multiple regression model,the coefficient of multiple determination will never decrease.

(True/False)

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Question 59

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: $\begin{array}{clll}\underline{\text { Employee }} & \underline{Y(\$)} & \underline{X}_{1} &\underline{X}_{2}\\1 & 10 & 3 & 0 \\2 & 12 & 1 & 5 \\3 & 15 & 8 & 1 \\4 & 17 & 5 & 8 \\5 & 20 & 7 & 12 \\6 & 25 & 10 & 9\end{array}$ -Referring to Instruction 13-2,suppose an employee had never taken an economics course and managed to score a 5 on his performance rating.What is his estimated expected wage rate?

(Multiple Choice)

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Question 60

The total sum of squares (SST)in a regression model will never exceed the regression sum of squares (SSR).

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

Consider a regression in which b₂ = -1.5 and the standard error of this coefficient equals 0.3.To determine whether X₂ is a significant explanatory variable,you would compute an observed t-value of -5.0.

When a dummy variable is included in a multiple regression model,the interpretation of the estimated slope coefficient does not make any sense anymore.

When an additional explanatory variable is introduced into a multiple regression model,the coefficient of multiple determination will never decrease.

Introduction and Data Collection

Presenting Data in Tables and Charts

Numerical Descriptive Measures

Basic Probability

Some Important Discrete Probability Distributions

The Normal Distribution and Other Continuous Distributions

Sampling Distributions

Confidence Interval Estimation

Fundamentals of Hypothesis Testing: One-Sample Tests

Hypothesis Testing: Two Sample Tests

Analysis of Variance

Simple Linear Regression

Time-Series Forecasting and Index Numbers

Chi-Square Tests

Multiple Regression Model Building

Decision Making

Statistical Applications in Quality and Productivity Management

Further Non-Parametric Tests

Filters

Exam 13: Introduction to Multiple Regression

The total sum of squares (SST)in a regression model will never exceed the regression sum of squares (SSR).

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

Consider a regression in which b2 = -1.5 and the standard error of this coefficient equals 0.3.To determine whether X2 is a significant explanatory variable,you would compute an observed t-value of -5.0.

When a dummy variable is included in a multiple regression model,the interpretation of the estimated slope coefficient does not make any sense anymore.

When an additional explanatory variable is introduced into a multiple regression model,the coefficient of multiple determination will never decrease.

Introduction and Data Collection

Presenting Data in Tables and Charts

Numerical Descriptive Measures

Basic Probability

Some Important Discrete Probability Distributions

The Normal Distribution and Other Continuous Distributions

Sampling Distributions

Confidence Interval Estimation

Fundamentals of Hypothesis Testing: One-Sample Tests

Hypothesis Testing: Two Sample Tests

Analysis of Variance

Simple Linear Regression

Time-Series Forecasting and Index Numbers

Chi-Square Tests

Multiple Regression Model Building

Decision Making

Statistical Applications in Quality and Productivity Management

Further Non-Parametric Tests

Filters

Consider a regression in which b₂ = -1.5 and the standard error of this coefficient equals 0.3.To determine whether X₂ is a significant explanatory variable,you would compute an observed t-value of -5.0.