Exam 19: Multiple Regression

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:

The most commonly used method to remedy non-normality or heteroscedasticity in regression analysis is to transform the dependent variable. The most commonly used transformations are $y ^ { \prime } = \log y ( \text { provided } y \geq 0 )$ , $y ^ { \prime } = y ^ { 2 }$ , $y ^ { \prime } = \sqrt { y } ( \text { provided } y \geq 0 )$ , and $y ^ { \prime } = 1 / y$ .

Multiple linear regression is used to estimate the linear relationship between one dependent variable and more than one independent variables.

In a multiple regression model, the error variable $\varepsilon$ is assumed to have a mean of:

For the estimated multiple regression model $\hat{y}$ = 30 - 4x₁ + 5x₂ +3 x₃, a one unit increase in x₃, holding x₁ and x₂ constant, will result in which of the following changes in y?

In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is $= 72 + 3.2 x _ { 1 } + 1.5 x _ { 2 } - x _ { 3 }$ . For this model, SST = 800 and SSE = 245. The value of the F-statistic for testing the significance of this model is 15.102.

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

A multiple regression model involves8 independent variables and 32 observations. If we want to test at the 5% significance level the parameter $\beta _ { 4 }$ , the critical value will be:

In a multiple regression analysis involving 50 observations and 5 independent variables, SST = 475 and SSE = 71.25. The multiple coefficient of determination is 0.85.

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 1% significance level to conclude that the final mark and the mid-term mark are positively 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 What is the coefficient of determination? What does this statistic tell you?

In a regression model involving 60 observations, the following estimated regression model was obtained: $= 51.4 + 0.70 x _ { 1 } + 0.679 x _ { 2 } - 0.378 x _ { 3 }$ For this model, total variation in y = SSY = 119,724 and SSR = 29,029.72. The value of MSE is:

For the multiple regression model $= 75 + 25 x _ { 1 } - 15 x _ { 2 } + 10 x _ { 3 }$ , if $x _ { 2 }$ were to increase by 5, holding $x _ { 1 }$ and $x _ { 3 }$ constant, the value of y would:

In a multiple regression analysis, if the model provides a poor fit, this indicates that:

Pop-up coffee vendors have been popular in the city of Adelaide in 2013. A Pop-up coffee 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 a. Write down the multiple regression model. b. Interpret the coefficient of Temperature. c. Interpret the coefficient of Pastries/biscuits.

In a multiple regression analysis involving 25 data points and 5 independent variables, the sum of squares terms are calculated as: total variation in y = SSY = 500, SSR = 300, and SSE = 200. In testing the validity of the regression model, the F-value of the test statistic will be:

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 the overall model's validity at the 5% significance level.

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 10% significance level to determine whether annual household income and annual family clothes expenditure are linearly related.

In multiple regression models, the values of the error variable $\varepsilon$ are assumed to be:

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.