Exam 19: Multiple Regression

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 SUMMARV OUTPUT Regression Stotistics Multiple R 0.87 R Square 0.75 Adjusted R Square 0.59 Standard Error 5.95 Otservations 6.00 ANOVA Significonce df SS MS F F Regression 2.00 322.14 161.07 4.55 0.12 Residual 3.00 106.20 35.40 Total 5.00 428.33 Coefficients Standard Error tStot Pvolue Lower 95\% Upper 95\% Intercept 18.68 37.8 0.49 0.66 -101.88 139.24 Temperature -0.50 0.83 -0.60 0.59 -3.15 2.15 Patries/bisouits 0.49 2.02 0.24 0.82 -5.94 6.92 Test the significance of the coefficient on Pasties/biscuits against a two tailed alternative. Use the 5% level of significance.

Which of the following measures can be used to assess a multiple regression model's fit?

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 $y =$ $1.74 + 0.091 x _ { 1 } + 0.93 x _ { 2 } + 0.26 x _ { 3 }$ 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 S = 2.06 R-Sq = 59.6%. ANALYSIS OF VARIANCE Source of Variation Regression 3 288 96 22.647 Error 46 195 4.239 Total 49 483 Interpret the coefficient $b _ { 2 }$ .

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 SUMMARV OUTPUT Regression Stotistics Multiple R 0.87 R Square 0.75 Adjusted R Square 0.59 Standard Error 5.95 Otservations 6.00 ANOVA Significonce df SS MS F F Regression 2.00 322.14 161.07 4.55 0.12 Residual 3.00 106.20 35.40 Total 5.00 428.33 Coefficients Standard Error tStot Pvolue Lower 95\% Upper 95\% Intercept 18.68 37.8 0.49 0.66 -101.88 139.24 Temperature -0.50 0.83 -0.60 0.59 -3.15 2.15 Patries/bisouits 0.49 2.02 0.24 0.82 -5.94 6.92 Test the significance of the overall regression model, at α of 5%.

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.

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

In a multiple regression model, the mean of the probability distribution of the error variable $\varepsilon$ is assumed to 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 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 R-Sq = 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 to conclude at the 5% significance level that the final mark and the number of skipped lectures are linearly related?

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 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 R-Sq = 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?

For each x term in the multiple regression equation, the corresponding $\beta$ is referred to as a partial regression coefficient or slope of the independent variable.

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 $y =$ $1.74 + 0.091 x _ { 1 } + 0.93 x _ { 2 } + 0.26 x _ { 3 }$ 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 S = 2.06 R-Sq = 59.6%. ANALYSIS OF VARIANCE Source of Variation Regression 3 288 96 22.647 Error 46 195 4.239 Total 49 483 What is the coefficient of determination? What does this statistic tell you?

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:

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 $y =$ $1.74 + 0.091 x _ { 1 } + 0.93 x _ { 2 } + 0.26 x _ { 3 }$ 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 S = 2.06 R-Sq = 59.6%. ANALYSIS OF VARIANCE Source of Variation 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.

In order to test the significance of a multiple regression model involving 4 independent variables and 30 observations, the number of degrees of freedom for the numerator and denominator for the critical value of F are 4 and 26, respectively.

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 age are linearly related?

In multiple regression, the Durbin-Watson test is used to determine if there is autocorrelation in the regression model

Which of the following best describes a multiple linear regression model?

The problem of multicollinearity arises when the:

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

When the independent variables are correlated with one another in a multiple regression analysis, this condition is called: