Exam 19: Multiple Regression

In multiple regression analysis involving 9 independent variables and 110 observations, the critical value of t for testing individual coefficients in the model will have:

If none of the data points for a multiple regression model with two independent variables were on the regression plane, then the multiple coefficient of determination would be:

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 $= 41.6 - 3.18 x _ { 1 } - 1.17 x _ { 2 } + .63 x _ { 3 }$ Predictor Coef StDev 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 $\mathrm { S } = 13.74 \quad \mathrm { R } - \mathrm { Sq } = 30.0 \%$ ANALYSIS OF VARIANCE Source of Variation 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?

In a multiple regression model, the following statistics are given: SSE = 100, $R ^ { 2 } = 0.995$ , k = 5, n = 15. The multiple coefficient of determination adjusted for degrees of freedom is:

The graphical depiction of the equation of a multiple regression model with k independent variables (k > 1) is referred to as:

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

Pop-up coffee vendors have been popular in the city of Adelaide in 2013. A vendor is interested in knowing how temperature (in degrees Celsius) and number of different pastries and biscuits offered to customers impacts daily hot coffee sales revenue (in $00's). A random sample of 6 days was taken, with the daily hot coffee sales revenue and the corresponding temperature and number of different pastries and biscuits offered on that day, noted. Excel output for a multiple linear regression is given below: Coffee sales revenue Temperature Pastries/biscuits 6.5 25 7 10 17 13 5.5 30 5 4.5 35 6 3.5 40 3 28 9 15 Comment on the difference between the coefficient of determination and the Adjusted coefficient of determination.

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 regression is:

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation $H _ { 1 } :$ There is negative first-order autocorrelation, given that the Durbin-Watson statistic d = 2.50, n = 40, k = 3 and $\alpha =$ 0.05.

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 ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $X _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below: THE REGRESSION EQUATION IS $y =$ $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ Predictor Coef StDev 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 R-Sq = 22.5%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 936 312 3.477 Error 36 3230 89.722 Total 39 4166 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 regression analysis, the total variation in the dependent variable y, measured by $\sum \left( y _ { i } - \bar { y } \right) ^ { 2 }$ , can be decomposed into two parts: the explained variation, measured by SSR, and the unexplained variation, measured by SSE.

In a regression model involving 30 observations, the following estimated regression model was obtained: $= 60 + 2.8 x _ { 1 } + 1.2 x _ { 2 } - x _ { 3 }$ . For this model, total variation in y = SSY = 800 and SSE = 200. The value of the F-statistic for testing the validity of this model is:

In a multiple regression, a large value of the test statistic F indicates that most of the variation in y is explained by the regression equation, and that the model is useful; while a small value of F indicates that most of the variation in y is unexplained by the regression equation, and that the model is useless.

A multiple regression analysis involving 3 independent variables and 25 data points results in a value of 0.769 for the unadjusted multiple coefficient of determination. The adjusted multiple coefficient of determination is:

In multiple regression, 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 $R ^ { 2 }$ = 1.

In a multiple regression analysis, when there is no linear relationship between each of the independent variables and the dependent variable, then:

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.

For a multiple regression model:

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 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 $y =$ $22.3 + 0.41 x _ { 1 } - 0.29 x _ { 2 } - 0.12 x _ { 3 }$ 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 S = 4.51 R-Sq = 34.8%. ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 227 75.667 3.730 Error 21 426 20.286 Total 24 653 Is there sufficient evidence at the 1% significance level to indicate that hours of television watched and education are negatively linearly related?