Exam 17: Multiple Regression

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 Coeff St. Error -value Intercept -1.6335 5.807 -0.281 0.7798 Family Incame 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 -{Real Estate Builder Narrative} What minimum annual income would an individual with a family size of 9 and 10 years of education need to attain a predicted 5,000 square foot home?

A high correlation between two independent variables is an indication of ____________________.

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

The range of the values of the Durbin-Watson statistic d is:

In a multiple regression analysis involving 40 observations and 5 independent variables,the following statistics are given: Total variation in y = 350 and SSE = 50.Then,the coefficient of determination is:

The coefficient of determination R² measures the proportion of variation in y that is explained by the explanatory variables included in the model.

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 $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + €$ ,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 $\tilde { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predicter Coef StDsv 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 Repressian 3 3716 1238.667 6.558 Esrar 46 8688 188.870 Total 49 12404 -{Student's Final Grade Narrative} Does this data provide enough evidence to conclude at the 5% significance level that the model is useful in predicting the final grade?

Large values of the Durbin-Watson statistic d (d > 2)indicate a positive first-order autocorrelation.

When there is more than one independent variable in a regression model,we refer to the graphical depiction of the equation as a(n)____________________ rather than as a straight line.

If the Durbin-Watson statistic has a value close to 4,which assumption is violated?

A multiple regression model has:

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 Coeff St. Error -value Intercept -1.6335 5.807 -0.281 0.7798 Family Incame 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 -{Real Estate Builder Narrative} Which of the independent variables in the model are significant at the 2% level?

The computer output for the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \sigma$ is shown below.However,because of a printer malfunction some of the results are not shown.These are indicated by the boldface letters a to i.Fill in the missing results (up to three decimal places). Fredictar Coef Constant 4.11 3.51 1.25 -0.71 0.30 $S = d \quad \text { R-Sq } = e$ ANALYSIS OF VARIANCE Source of Variation Regressian 2 412 Error 37 Total 39 974

For a multiple regression model,the total variation in y can be expressed as:

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

The validity of a multiple regression model is tested using a(n)_________ test.

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 Coeff St. Error -value Intercept -1.6335 5.807 -0.281 0.7798 Family Incame 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 -{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?

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 Coeff St. Error -value Intercept -1.6335 5.807 -0.281 0.7798 Family Incame 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 -{Real Estate Builder Narrative} What are the regression degrees of freedom that are missing from the output?

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 $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + €$ ,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 $\tilde { y } = 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predicter Coef StDsv 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 Repressian 3 3716 1238.667 6.558 Esrar 46 8688 188.870 Total 49 12404 -{Student's Final Grade Narrative} Interpret the coefficient b₁.

In reference to the equation $\tilde { y } = 1.86 - 0.51 x _ { 1 } + 0.60 x _ { 2 }$ ,the value 0.60 is the average change in y per unit change in x₂,regardless of the value of x₁.