Exam 16: Multiple Regression

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 model, the mean of the probability distribution of the error variable $\varepsilon$ is assumed to be:

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 $y ^ { \prime } = 1 / y$ .

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 education are negatively linearly related?

If all the points for a multiple regression model with two independent variables were on the regression plane, then the coefficient of determination would equal:

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

For a multiple regression model, the following statistics are given: Total variation in y = SST = 250, SSE = 50, k = 4, n = 20. The coefficient of determination adjusted for degrees of freedom is:

In a multiple regression analysis, there are 20 data points and 4 independent variables, and the sum of the squared differences between observed and predicted values of y is 180. The multiple standard error of estimate will be:

Consider the following statistics of a multiple regression model: n = 30 k = 4 SST = 1500 SSE = 260. a. Determine the standard error of estimate. b. Determine the coefficient of determination. c. Determine the F-statistic.

A multiple regression analysis involving 3 independent variables and 25 data points results in a value of 0.769 for the (unadjusted) coefficient of determination. The adjusted coefficient of determination 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 ratio MSR/MSE in a multiple linear regression model?

The computer output for the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \varepsilon$ is shown below. However, because of a printer malfunction some of the results are not shown. These are indicated by the boldface letters a to i. Fill in the missing results (up to three decimal places). Predictor Coef StDev T Constant 0.120 3.18 0.068 3.38 0.024 0.010 se = d R2 = e. ANALYSIS OF VARIANCE Source of Variation Regression 2 7.382 Error 22 f h Total 24 7.530

In reference to the equation $\hat { y }$ = 1.86 - 0.51x1 + 0.60 x2, the value 0.60 is the change in $y$ per unit change in $x _ { 2 }$ , regardless of the value of $x _ { 1 }$ .

Given the multiple linear regression equation, $\hat { y }$ = -0.80 + 0.12 x1 + 0.08 x2, the value -0.80 is the $y$ intercept.

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 Interpret the coefficients $\hat { \beta } _ { 1 }$ and $\hat { \beta } _ { 3 }$ .

In a multiple regression model, the standard deviation of the error variable $\varepsilon$ is assumed to be:

In a regression model involving 50 observations, the following estimated regression model was obtained: ŷ = 10.5 + 3.2x1 + 5.8x2 + 6.5x3. For this model, SSR = 450 and SSE = 175. The value of MSE 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 cholesterol level and the age at death are negatively linearly related?

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