Exam 16: Multiple Regression

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 ŷ = $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ Predictor Coef StDev 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 se = 9.47 R2 = 22.5%. ANALYSIS OF VARIANCE Source of Variation SS MS F Regression 3 936 312 3.477 Error 36 3230 89.722 Total 39 4166 Is there enough evidence at the 1% significance level to infer that the average number of hours of exercise per week and the age at death are linearly related?

In a multiple regression model, the following statistics are given: SSE = 100, R2 = 0.995, k = 5, n = 15. The coefficient of determination adjusted for degrees of freedom is:

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 SUMMARY OUTPUT Regression Statistios Multiple R 0.87 R Square 0.75 Adjusted R Square 0.59 Standard Error 5.95 Observations 6.00 ANOVA df SS MS F Significance F Regression 2.00 322.14 161.07 4.55 0.12 Residual 3.00 106.20 35.40 Total 5.00 428.33 Coeffients Standard Error tStat P-value Lower 95\% Upper 95\% Intercept 18.68 37.88 0.49 0.66 -101.88 139.24 Temperature -0.50 0.83 -0.60 0.59 -3.15 2.15 Pastries/biscuits 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.

An estimated multiple regression model has the form $\hat { y } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x _ { 1 } + \hat { \beta } _ { 2 } x _ { 2 } .$ Which of the following best describes $\hat { \beta } _ { 2 }$ ?

In testing the significance of a multiple regression model in which there are three independent variables, the null hypothesis is Ho: β0 = β1 = β2 = β3.

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

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 to conclude at the 5% significance level that the model is useful in predicting the final mark?

The problem of multicollinearity arises when the:

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

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

In multiple regression, the descriptor 'multiple' refers to more than one independent variable.

In a multiple regression analysis involving 6 independent variables and a sample of 19 data points the total variation in y is SST = 900 and the amount of variation in y that is explained by the variations in the independent variables is SSR = 600. The value of the F-test statistic for this model is:

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 ŷ = $55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$ Predictor Coef StDev 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 se = 9.47 R2 = 22.5%. ANALYSIS OF VARIANCE Source of Variation 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 number of points by which the individual's blood pressure exceeded the recommended value and the age at death are negatively linearly related?

The adjusted coefficient of determination is adjusted for the number of independent variables and the sample size.

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

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:

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 = SST = 500, SSR = 300, and SSE = 200. In testing the validity of the regression model, the F-value of the test statistic will be:

In a multiple regression analysis involving 30 data points, the standard error of estimate squared is calculated as s2 = 1.5 and the sum of squares for error as SSE = 36. The number of the independent variables must be:

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.

Given the multiple linear regression equation, $\hat { y } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x _ { 1 } + \hat { \beta } _ { 2 } x _ { 2 } ,$ the value of $\hat { \beta } _ { 2 }$ is the estimated average increase in y for a one unit increase in x2, whilst holding x1 constant.