Exam 16: Multiple Regression

In testing the validity of a multiple regression model involving 5 independent variables and 30 observations, the numbers of degrees of freedom for the numerator and denominator (respectively) for the critical value of F will 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 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 final mark and the number of skipped lectures are linearly related?

Which of the following best explains a small F-statistic when testing the validity of a multiple regression model?

For the multiple regression model $\hat { y }$ = 75 + 25x1 - 15 x2 + 10 x3, if $x _ { 2 }$ were to increase by 5, holding $x _ { 1 }$ and $x _ { 3 }$ constant, the value of y would:

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

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

Excel prints a second $R ^ { 2 }$ statistic, called the coefficient of determination adjusted for degrees of freedom, which has been adjusted to take into account the sample size and the number of independent variables.

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

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 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 a. Write down the multiple regression model. b. Interpret the coefficient of Temperature. c. Interpret the coefficient of Pastries/biscuits.

Which of the following best describes first-order autocorrelation?

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 Comment on the difference between the coefficient of determination and the Adjusted coefficient of determination.

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 Test the overall validity of the model at the 5% significance level.

A multiple regression model has the form $\hat { y } = \hat { \beta } _ { 0 } + \hat { \beta } _ { 1 } x _ { 1 } + \hat { \beta } _ { 2 } x _ { 2 } .$ The coefficient $\hat { \beta } _ { 2 }$ is interpreted as the change in $y$ per unit change in x2.

For a multiple regression model with n = 35 and k = 4, the following statistics are given: Total variation in y = SST = 500 and SSE = 100. The coefficient of determination is:

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

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 Is there sufficient evidence at the 1% significance level to indicate that hours of television watched and age are linearly related?

Given the following statistics of a multiple regression model, can we conclude at the 5% significance level that $x _ { 1 }$ and y are linearly related? n = 42, k = 6, $\hat { \beta } _ { 1 }$ = -5.30 $S _ {\hat { \beta } _ { 1 }}$ = 1.5

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

In multiple regression with k independent variables, the t-tests of the individual coefficients allow us to determine whether $\beta _ { i } \neq 0$ (for i = 1, 2, …, k), which tells us whether a linear relationship exists between $x _ { i }$ and y.

In a regression model involving 60 observations, the following estimated regression model was obtained: $\hat { y }$ = 51.4 + 0.70x1 + 0.679 x2 - 0.378 x3. For this model, total variation in y = SST = 119,724 and SSR = 29,029.72. The value of MSE is: