Exam 16: Multiple Regression

A statistician wanted to determine whether the demographic variables of age, education and income influence the number of hours of television watched per week. A random sample of 25 adults was selected to estimate the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . Where: y = number of hours of television watched last week. $x _ { 1 }$ = age. $x _ { 2 }$ = number of years of education. $x _ { 3 }$ = income (in $1000s). The computer output is shown below. THE REGRESSION EQUATION IS ŷ = $22.3 + 0.41 x _ { 1 } - 0.29 x _ { 2 } - 0.12 x _ { 3 }$ Predictor Coef StDev Constant 22.3 10.7 2.084 0.41 0.19 2.158 -0.29 0.13 -2.231 -0.12 0.03 -4.00 se = 4.51 R2 = 34.8% ANALYSIS OF VARIANCE Source of Variation SS MS F Regression 3 227 75.667 3.730 Error 21 426 20.286 Total 24 653 What is the coefficient of determination? What does this statistic tell you?

An economist wanted to develop a multiple regression model to enable him to predict the annual family expenditure on clothes. After some consideration, he developed the multiple regression model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . Where: y = annual family clothes expenditure (in $1000s) $x _ { 1 }$ = annual household income (in $1000s) $x _ { 2 }$ = number of family members $x _ { 3 }$ = number of children under 10 years of age The computer output is shown below. THE REGRESSION EQUATION IS ŷ = $1.74 + 0.091 x _ { 1 } + 0.93 x _ { 2 } + 0.26 x _ { 3 }$ Predictor Coef StDev Constant 1.74 0.630 2.762 0.091 0.025 3.640 0.93 0.290 3.207 0.26 0.180 1.444 se = 2.06, R2 = 59.6%. ANALYSIS OF VARIANCE Source of Variation SS MS F Regression 3 288 96 22.647 Error 46 195 4.239 Total 49 483 Test at the 1% significance level to determine whether the number of family members and annual family clothes expenditure are linearly related.

A multiple regression model involves 40 observations and 4 independent variables produces SST = 100 000 and SSR = 82,500. The value of MSE is 500.

A multiple regression model involves 5 independent variables and the sample size is 30. If we want to test the validity of the model at the 5% significance level, the critical value is:

For the multiple regression model $\ddot { y } = 40 + 15 x _ { 1 } - 10 x _ { 2 } + 5 x _ { 3 }$ , if $x _ { 2 }$ were to increase by 5 units, holding $x _ { 1 }$ and $x _ { 3 }$ constant, the value of $y$ would decrease by 50 units, on average.

An estimated multiple regression model has the form ŷ = 100 − 2x1 + 9x2. As x1 increases by 1 unit while holding x2 constant, which of the following best describes the change in y?

If multicollinearity exists among the independent variables included in a multiple regression model, then:

A multiple regression equation includes 5 independent variables, and the coefficient of determination is 0.64. The percentage of the variation in y that is explained by the regression equation is:

A multiple regression analysis that includes 25 data points and 4 independent variables produces SST = 400 and SSR = 300. The multiple standard error of estimate will be 5.

An estimated multiple regression model has the form ŷ = 8 + 3x1  5x2  4x3. As x1 increases by 1 unit, with x2 and x3 held constant, the value of y, on average, is estimated to:

A multiple regression analysis that includes 20 data points and 4 independent variables results in total variation in y = SST = 200 and SSR = 160. The multiple standard error of estimate will be:

Which of the following statements is not true?

A statistics professor investigated some of the factors that affect an individual student's final grade in his or her course. He proposed the multiple regression model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . Where: y = final mark (out of 100). $x _ { 1 }$ = number of lectures skipped. $x _ { 2 }$ = number of late assignments. $x _ { 3 }$ = mid-term test mark (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below. THE REGRESSION EQUATION IS ŷ = $41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predictor Coef StDev 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 se = 13.74, R2 = 30.0%. ANALYSIS OF VARIANCE Source of Variation SS MS F Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 What is the coefficient of determination? What does this statistic tell you?

Which of the following best describes the Durbin-Watson test?

In a multiple regression analysis involving 20 observations and 5 independent variables, total variation in y = SST = 250 and SSE = 35. The coefficient of determination adjusted for degrees of freedom is:

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.

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation HA : There is positive first-order autocorrelation, given that: the Durbin-Watson statistic d = 0.686, n = 16, k = 1 and $\alpha =$ 0.05.

A statistics professor investigated some of the factors that affect an individual student's final grade in his or her course. He proposed the multiple regression model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ . Where: y = final mark (out of 100). $x _ { 1 }$ = number of lectures skipped. $x _ { 2 }$ = number of late assignments. $x _ { 3 }$ = mid-term test mark (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below. THE REGRESSION EQUATION IS ŷ = $41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predictor Coef StDev 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 se = 13.74, R2 = 30.0%.. ANALYSIS OF VARIANCE Source of Variation SS MS F Regression 3 3716 1238.667 6.558 Error 46 8688 188.870 Total 49 12404 Do these data provide enough evidence at the 5% significance level to conclude that the final mark and the number of late assignments are negatively linearly related?

The adjusted coefficient of determination is adjusted for the:

A multiple regression the coefficient of determination is 0.81. The percentage of the variation in $y$ that is explained by the regression equation is 81%.