Exam 17: Multiple Regression

If the Durbin-Watson statistic, d, has values greater than 2, this indicates

If a group of independent variables are not significant individually but are significant as a group at a specified level of significance, this is most likely due to:

The problem of multicollinearity arises when the:

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} Does this data provide enough evidence to conclude at the 5% significance level that the final grade and the number of skipped lectures are linearly related?

There are several clues to the presence of multicollinearity. One clue is when a regression coefficient exhibits the wrong ____________________.

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 b₁.

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 b₃.

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 percentage of the variability in house size is explained by this model?

In multiple regression, the standard error of estimate is defined by $s _ { z } = \sqrt { \operatorname { SSE } / ( n - k ) }$ , where n is the sample size and k is the number of independent 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} Which of the following values for the level of significance is the smallest for which at least two explanatory variables are significant individually: $\alpha$ = .01, .05, .10, and .15?

Discuss two indicators that can be found in an analysis that suggest multicollinearity is present.

Consider the following statistics of a multiple regression model: Total variation in y = 1000, SSE = 300, n = 50, and k = 4. a.Determine the standard error of estimate. b.Determine the coefficient of determination. c.Determine the F-statistic.

An adverse effect of multicollinearity is that the estimated regression coefficients of the independent variables that are correlated tend to have large sampling ____________________.

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} What is the adjusted coefficient of determination in this situation? What does this statistic tell you?

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

The range of the values of the Durbin-Watson statistic, d, is 0 $\le$ d $\le$ 4.

In a multiple regression analysis involving 4 independent variables and 30 data points, the number of degrees of freedom associated with the sum of squares for error, SSE, is 25.

A(n) ____________________ value of the F-test statistic indicates that the multiple regression model is valid.