Exam 17: Multiple Regression

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 minimum annual income would an individual with a family size of 4 and 16 years of education need to attain a predicted 10,000 square foot home?

The computer output for the multiple regression model $y : \beta _ { 0 }+ \beta _ { 1 }x_ { 1 }+ \beta _ { 2 }x_ { 2 }+ε$ 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). predictor 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 Repression 2 412 Error 37 Total 39 974

In a multiple regression analysis, there are 20 data points and 4 independent variables, and the sum of the squared differences between observed and predicted values of y is 180. The standard error of estimate will be:

If the value of the Durbin-Watson test statistic, d, satisfies the inequality d > 4-d_L, we conclude that positive first-order autocorrelation exists.

The adjusted coefficient of determination is adjusted for the:

We test an individual coefficient in a multiple regression model using a(n) _________ test.

A multiple regression model has:

In a multiple regression analysis involving 50 observations and 5 independent variables, the total variation in y is 475 and SSE = 71.25. Then, the coefficient of determination is 0.85.

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 $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 feet. What is the residual (in hundreds of square feet) for this data point?

A multiple regression model has the form $\hat { y } = b _ { 0 } + b _ { 1 } x _ { 1 } + b _ { 2 } x _ { 2 }$ . The coefficient b₁ is interpreted as the change in the average value of y per unit change in ________ holding ________ constant.

A multiple regression model involves 10 independent variables and 30 observations. If we want to test at the 5% significance level whether one of the coefficients is = 0 (vs. $\neq$ 0) the critical value will be:

If the value of the Durbin-Watson statistic, d, satisfies the inequality d_L $\le$ d $\le$ d_U, where d_L and d_U are the critical values for d, then the test for positive first-order autocorrelation is inconclusive.

To use the Durbin-Watson test to test for negative first-order autocorrelation, the null hypothesis will be H₀: ____________________ (there is/there is no) first-order autocorrelation.

In a multiple regression analysis involving k independent variables and n data points, the number of degrees of freedom associated with the sum of squares for error is:

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 model is useful in predicting the final grade?

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

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

One of the consequences of multicollinearity in multiple regression is inflated standard errors in some or all of the estimated slope coefficients.

The Durbin-Watson d statistic is used to check the assumption of normality.

If the Durbin-Watson statistic has a value close to 0, which assumption is violated?