Exam 17: Multiple Regression

Given that the Durbin-Watson test is conducted to test for positive first-order autocorrelation with $\alpha$ = .05, n = 20, and there are two independent variables in the model, the critical values for the test are d_L = __________ and d_U = __________, respectively.

A multiple regression model is assessed to be poor if the error sum of squares SSE and the standard error of estimate s _{$\varepsilon$} are both large, the coefficient of determination R² is close to 0, and the value of the test statistic F is large.

NARRBEGIN: Real Estate Builder Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below.SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R. Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA df SS MS F Signif F Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff st.error -value Intercept -1.6335 5.807 -0.281 0.798 Family Income 0.4485 0.1137 3.9545 0.0003 Family Size 4.2615 0.8062 5.286 0.0001 Education -0.6517 0.4319 -1.509 0.1383 NARREND -{Real Estate Builder Narrative} What are the residual degrees of freedom that are missing from the output?

NARRBEGIN: Real Estate Builder Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below.SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R. Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA df SS MS F Signif F Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff st.error -value Intercept -1.6335 5.807 -0.281 0.798 Family Income 0.4485 0.1137 3.9545 0.0003 Family Size 4.2615 0.8062 5.286 0.0001 Education -0.6517 0.4319 -1.509 0.1383 NARREND -{Real Estate Builder Narrative} At the 0.01 level of significance, what conclusion should the builder draw regarding the inclusion of income in the regression model?

A multiple regression is called "multiple" because it has several explanatory variables.

NARRBEGIN: Real Estate Builder Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below.SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R. Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA df SS MS F Signif F Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff st.error -value Intercept -1.6335 5.807 -0.281 0.798 Family Income 0.4485 0.1137 3.9545 0.0003 Family Size 4.2615 0.8062 5.286 0.0001 Education -0.6517 0.4319 -1.509 0.1383 NARREND -{Real Estate Builder Narrative} One individual in the sample had an annual income of $10,000, a family size of 1, and an education of 8 years. This individual owned a home with an area of 1,000 square fee (House = 10.00). What is the residual (in hundreds of square feet) for this data point?

A multiple regression model is assessed to be good if the error sum of squares SSE and the standard error of estimate s _{$\varepsilon$} are both small, the coefficient of determination R² is close to 1, and the value of the test statistic F is large.

NARRBEGIN: Real Estate Builder Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below.SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R. Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA df SS MS F Signif F Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff st.error -value Intercept -1.6335 5.807 -0.281 0.798 Family Income 0.4485 0.1137 3.9545 0.0003 Family Size 4.2615 0.8062 5.286 0.0001 Education -0.6517 0.4319 -1.509 0.1383 NARREND -{Real Estate Builder Narrative} Interpret the value of the Adjusted R-Square.

NARRBEGIN: Student's Final Grade Student's Final Grade A statistics professor investigated some of the factors that affect an individual student's final grade in her course. She proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ , where y is the final grade (out of 100 points), x₁ is the number of lectures skipped, x₂ is the number of late assignments, and x₃ is the midterm exam score (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below.THE REGRESSION EQUATION IS $\hat { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ PredictOr Coef SuDsv T Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $S = 13.74 \quad R - S q = 30.0 \%$ ANALYSIS OF VARIANCE Source of Variation Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 NARREND -{Student's Final Grade Narrative} What is the coefficient of determination? What does this statistic tell you?

In order to test the validity of a multiple regression model involving 5 independent variables and 30 observations, the numerator and denominator degrees of freedom for the critical value of F are, respectively,

There are several clues to the presence of multicollinearity. One clue is when an independent variable is added or deleted, the regression coefficients for the other variables ____________________.

NARRBEGIN: Life Expectancy Life Expectancy An actuary wanted to develop a model to predict how long individuals will live. After consulting a number of physicians, she collected the age at death (y), the average number of hours of exercise per week (x₁), the cholesterol level (x₂), and the number of points that the individual's blood pressure exceeded the recommended value (x₃). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below.THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁ - 0.021x₂ -0.061x₃ predictor Coef SUDev T Constant 55.8 11.8 4.729 1.79 0.44 4.068 -0.021 0.011 -1.909 -0.016 0.014 -1.143 $S = 9.47 \quad R - S q = 22.5 \%$ ANALYSIS OF VARIANCE Source of Variation Repression 3 936 312 3.477 Error 36 3230 89.722 Tatol 39 4166 NARREND -{Life Expectancy Narrative} Is there enough evidence at the 5% significance level to infer that the cholesterol level and the age at death are negatively linearly related?

In a multiple regression model, the value of the coefficient of determination has to fall between

Multicollinearity is a situation in which two or more of the independent variables are highly correlated with each other.

NARRBEGIN: Student's Final Grade Student's Final Grade A statistics professor investigated some of the factors that affect an individual student's final grade in her course. She proposed the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ , where y is the final grade (out of 100 points), x₁ is the number of lectures skipped, x₂ is the number of late assignments, and x₃ is the midterm exam score (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below.THE REGRESSION EQUATION IS $\hat { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ PredictOr Coef SuDsv T Constant 41.6 17.8 2.337 -3.18 1.66 -1.916 -1.17 1.13 -1.035 0.63 0.13 4.846 $S = 13.74 \quad R - S q = 30.0 \%$ ANALYSIS OF VARIANCE Source of Variation Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 NARREND -{Student's Final Grade Narrative} Interpret the coefficient b₂.

NARRBEGIN: Real Estate Builder Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below.SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R. Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA df SS MS F Signif F Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff st.error -value Intercept -1.6335 5.807 -0.281 0.798 Family Income 0.4485 0.1137 3.9545 0.0003 Family Size 4.2615 0.8062 5.286 0.0001 Education -0.6517 0.4319 -1.509 0.1383 NARREND -{Real Estate Builder Narrative} Suppose the builder wants to test whether the coefficient on education is significantly different from 0. What is the value of the relevant t-statistic?

In multiple regression analysis, when the response surface (the graphical depiction of the regression equation) hits every single point, the sum of squares for error SSE = 0, the standard error of estimate s _{$\varepsilon$} = 0, and the coefficient of determination R² = 1.

A multiple regression analysis involving three independent variables and 25 data points results in a value of 0.769 for the unadjusted coefficient of determination. Then, the adjusted coefficient of determination is:

NARRBEGIN: Real Estate Builder Real Estate Builder A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below.SUMMARY OUTPUT Regression Statistics Multiple R 0.865 R. Square 0.748 Adjusted R Square 0.726 Standard Error 5.195 Observations 50 ANOVA df SS MS F Signif F Regression 3605.7736 901.4434 0.0001 Residual 1214.2264 26.9828 Total 49 4820.0000 Coeff st.error -value Intercept -1.6335 5.807 -0.281 0.798 Family Income 0.4485 0.1137 3.9545 0.0003 Family Size 4.2615 0.8062 5.286 0.0001 Education -0.6517 0.4319 -1.509 0.1383 NARREND -{Real Estate Builder Narrative} What is the value of the calculated F-test statistic that is missing from the output for testing whether the whole regression model is significant?

For a multiple regression model the following statistics are given: Total variation in y = 250, SSE = 50, k = 4, and n = 20. Then, the coefficient of determination adjusted for the degrees of freedom is: