Question 1

Inspection of the following table of correlation coefficients for variables in a multiple regression analysis reveals potential multicollinearity with variables ___________. $$\begin{array} { | c | r | c | r | c | c | r | } 
\hline & y & x _ { 1 } & x _ { 2 } & x _ { 3 } & x _ { 4 } & x _ { 5 } \
\hline y & 1 & & & & & \
\hline x _ { 1 } & - 0.0857 & 1 & & & & \
\hline x _ { 2 } & - 0.20246 & 0.868358 & 1 & & & \
\hline x _ { 3 } & - 0.22631 & - 0.10604 & - 0.14853 & 1 & & \
\hline x _ { 4 } & - 0.28175 & - 0.0685 & 0.41468 & - 0.14151 & 1 & \
\hline x _ { 5 } & 0.271105 & 0.150796 & 0.129388 & - 0.15243 & 0.00821 & 1 \
\hline
\end{array}$$

Accepted Answer

The variables $x_1$ and $x_2$ show potential multicollinearity due to their high correlation coefficient of 0.868358, indicating that they share a strong linear relationship.

Question 2

Inspection of the following table of t values for variables in a multiple regression analysis reveals that the first independent variable that will be entered into the regression model by the forward selection procedure will be ___________. $$\begin{array} { | c | r | c | r | c | c | r | } 
\hline & y & x _ { 1 } & x _ { 2 } & x _ { 3 } & x _ { 4 } & x _ { 5 } \
\hline y & 1 & & & & & \
\hline x _ { 1 } & - 0.0857 & 1 & & & & \
\hline x _ { 2 } & - 0.20246 & 0.868358 & 1 & & & \
\hline x _ { 3 } & - 0.22631 & - 0.10604 & - 0.14853 & 1 & & \
\hline x _ { 4 } & - 0.28175 & - 0.0685 & 0.41468 & - 0.14151 & 1 & \
\hline x _ { 5 } & 0.271105 & 0.150796 & 0.129388 & - 0.15243 & 0.00821 & 1 \
\hline
\end{array}$$

Accepted Answer

The answer of Inspection of the following table of t...

Question 3

An appropriate method to identify multicollinearity in a regression model is to ____.

Accepted Answer

A correlation matrix can help identify multicollinearity by showing high correlations between independent variables.

Question 4

Which of the following iterative search procedures for model-building in a multiple regression analysis adds variables to model as it proceeds, but does not reevaluate the contribution of previously entered variables?

Accepted Answer

Forward selection starts with no variables in the model and adds variables one at a time, based on their statistical significance, without reevaluating the contribution of variables that have already been included.

Question 5

Inspection of the following table of t values for variables in a multiple regression analysis reveals that the first independent variable that will be entered into the regression model by the forward selection procedure will be ___________. $$\begin{array} { | c | c | c | c | c | c | c | } 
\hline & y & x _ { 1 } & x _ { 2 } & x _ { 3 } & x _ { 4 } & x _ { 5 } \
\hline y & 1 & & & & & \
\hline x _ { 1 } & 0.854168 & 1 & & & & \
\hline x _ { 2 } & - 0.11828 & - 0.00383 & 1 & & & \
\hline x _ { 3 } & - 0.12003 & - 0.08499 & - 0.14523 & 1 & & \
\hline x _ { 4 } & 0.525901 & 0.118169 & - 0.14876 & 0.050042 & 1 & \
\hline x _ { 5 } & - 0.18105 & - 0.07371 & 0.995886 & - 0.14151 & - 0.16934 & 1 \
\hline
\end{array}$$

Accepted Answer

The first independent variable that will be entered into the regression model by the forward selection procedure is the one with the largest t-value. In this case, x1 has the largest t-value (0.854168), so it will be entered into the model first.

Question 6

Which of the following iterative search procedures for model-building in a multiple regression analysis starts with all independent variables in the model and then drops non-significant independent variables is a step-by-step manner?

Accepted Answer

Backward elimination starts with all independent variables in the model and then drops non-significant independent variables in a step-by-step manner.

Question 7

An acceptable method of managing multicollinearity in a regression model is the ___.

Accepted Answer

Stepwise regression is a method that can help manage multicollinearity by iteratively adding or removing predictors based on specific criteria, thus potentially reducing multicollinearity issues.

Question 8

An "all possible regressions" search of a data set containing 5 independent variables will produce ______ regressions.

Accepted Answer

An "all possible regressions" search involves examining every possible combination of independent variables, including the model with no independent variables (just the intercept) and each model with one or more of the independent variables. For 5 independent variables, the total number of combinations is $2^5 = 32$, but since one of these combinations includes no variables at all, the correct answer is $32 - 1 = 31$.

Question 9

Inspection of the following table of t values for variables in a multiple regression analysis reveals that the first independent variable entered by the forward selection procedure will be ___________. $$\begin{array} { | c | c | c | c | c | c | c | } 
\hline & y & x _ { 1 } & x _ { 2 } & x _ { 3 } & x _ { 4 } & x _ { 5 } \
\hline y & 1 & & & & & \
\hline x _ { 1 } & - 0.44008 & 1 & & & & \
\hline x _ { 2 } & 0.566053 & - 0.51728 & 1 & & & \
\hline x _ { 3 } & 0.064919 & - 0.22264 & - 0.00734 & 1 & & \
\hline x _ { 4 } & - 0.35711 & 0.028957 & - 0.49869 & 0.260586 & 1 & \
\hline x _ { 5 } & 0.426363 & - 0.20467 & 0.078916 & 0.207477 & 0.023839 & 1 \
\hline
\end{array}$$

Accepted Answer

The first independent variable entered by the forward selection procedure is typically the one with the highest correlation with the dependent variable (y). In the table, $x_2$ has the highest correlation with $y$ (0.566053), making it the first variable to be entered.

Question 10

An "all possible regressions" search of a data set containing "k" independent variables will produce __________ regressions.

Accepted Answer

An "all possible regressions" search involves considering every possible combination of the "k" independent variables, including the possibility of using none of them (which is not actually a regression but is counted in the total for completeness). The total number of combinations of "k" items is given by 2^k (since for each variable, there are two possibilities: either it is included in the model or it is not). However, since the scenario of using none of the variables is not considered a valid regression model, we subtract 1 from this total, resulting in 2^k - 1 possible regressions.

Question 11

Carlos Cavazos, Director of Human Resources, is exploring employee absenteeism at the Plano Piano Plant.A multiple regression analysis was performed using the following variables.The results are presented below. $$\begin{array} { | l | l | } 
\hline \text { Variable } & \text { Description } \
\hline Y & \text { number of days absent last fiscal year } \
\hline x _ { 1 } & \text { comrnuting distarnce (in miles) } \
\hline x _ { 2 } & \text { employee's age (in years) } \
\hline x _ { 3 } & \text { single-parent household } ( 0 = \text { no, } 1 = \text { yes } ) \
\hline x _ { 4 } & \text { length of employment at PpP (in years) } \
\hline x _ { 5 } & \text { shift } ( 0 = \text { day } 1 = \text { night) } \
\hline
\end{array}$$ $$\begin{array} { | c | c | c | c | c | } 
\hline & \text { Coefficients } & \text { Standard Error } & t \text { Statistic } & p \text {-value } \
\hline \text { Intercept } & 6.594146 & \mathbf { 3 . 2 7 3 0 0 5 } & \mathbf { 2 . 0 1 4 7 0 7 } & \mathbf { 0 . 0 4 7 6 7 1 } \
\hline \boldsymbol { x } _ { 1 } & - 0.18019 & 0.141949 & - 1.26939 & 0.208391 \
\hline \mathbf { x } _ { 2 } & 0.268156 & 0.260643 & 1.028828 & 0.307005 \
\hline \boldsymbol { x } _ { 3 } & - 2.31068 & 0.962056 & - 2.40182 & 0.018896 \
\hline \mathbf { x } _ { 4 } & - 0.50579 & 0.270872 & - 1.86725 & 0.065937 \
\hline \boldsymbol { x } _ { 5 } & \mathbf { 2 . 3 2 9 5 1 3 } & 0.940321 & 2.47736 & 0.015584 \
\hline
\end{array}$$ $$\begin{array} { | c | c | c | c | c | c | } 
\hline & \mathrm { df } & \text { SS } & \text { ME } & F & p \text {-value } \
\hline \text { Repression } & 5 & 279.358 & 55.8716 & 4.423755 & \mathbf { 0 . 0 0 1 5 3 2 } \
\hline \text { Residual } & 67 & 846.2036 & 12.6299 & & \
\hline \text { Total } & 72 & 1125.562 & & & \
\hline
\end{array}$$ $$\begin{array} { | c | c | c | } 
\hline R = 0.498191 & R ^ { 2 } = 0.248194 & \text { Adj } R ^ { 2 } = 0.192089 \
\hline \mathrm { s } _ { \mathrm { e } } = 3.553858 & n = 73 & \
\hline
\end{array}$$ Which of the following conclusions can be drawn from the above results?

Accepted Answer

The answer of Carlos Cavazos, Director of Human Resources, is...

Question 12

An "all possible regressions" search of a data set containing 8 independent variables will produce ______ regressions.

Accepted Answer

An "all possible regressions" search involves considering every possible combination of the independent variables, including the possibility of using none of them and each of them individually. For 8 independent variables, the total number of combinations is 2^8 (since each variable can either be included or not), which equals 256. However, since one of these combinations includes using none of the variables, which isn't a regression, the actual number of regressions is 256 - 1 = 255.

Question 13

Suppose a company is interested in understanding the effect of age and sex on the likelihood a customer will purchase a new product.The data analyst intends to run a logistic regression on her data.Which of the following variable(s)will the analyst need to code as 0 or 1 prior to performing the logistic regression analysis?

Accepted Answer

The answer of Suppose a company is interested in understanding...

Question 14

Inspection of the following table of t values for variables in a multiple regression analysis reveals that the first independent variable entered by the forward selection procedure will be ___________. $$\begin{array} { | l | r | r | r | r | r | r | } 
\hline & y & x _ { 1 } & x _ { 2 } & x _ { 3 } & x _ { 4 } & x _ { 5 } \
\hline y & 1 & & & & & \
\hline x _ { 1 } & - 0.1661 & 1 & & & & \
\hline x _ { 2 } & 0.231849 & - 0.51728 & 1 & & & \
\hline x _ { 3 } & 0.423522 & - 0.22264 & - 0.00734 & 1 & & \
\hline x _ { 4 } & - 0.33227 & 0.028957 & - 0.49869 & 0.260586 & 1 & \
\hline x _ { 5 } & 0.199796 & - 0.20467 & 0.078916 & 0.207477 & 0.023839 & 1 \
\hline
\end{array}$$

Accepted Answer

The first independent variable entered by the forward selection procedure will be the one with the highest correlation with the dependent variable y, which is $x_3$ with a correlation of 0.423522.

Question 15

An "all possible regressions" search of a data set containing 7 independent variables will produce ______ regressions.

Accepted Answer

An "all possible regressions" search involves examining every possible combination of independent variables, including the cases where some variables are excluded. For 7 independent variables, the total number of combinations is 2^7 - 1 (subtracting 1 because the case where no variables are included is not considered a regression). This equals 128 - 1 = 127 regressions.

Question 16

Large correlations between two or more independent variables in a multiple regression model could result in the problem of ________.

Accepted Answer

Multicollinearity occurs when two or more independent variables in a regression model are highly correlated, making it difficult to determine the individual effect of each variable.

Question 17

Which of the following iterative search procedures for model-building in a multiple regression analysis reevaluates the contribution of variables previously include in the model after entering a new independent variable?

Accepted Answer

Stepwise regression both adds and removes predictors as needed in each step, reevaluating the model each time a new variable is added to ensure that all included variables contribute significantly to the model's predictive power.

Question 18

A useful technique in controlling multicollinearity involves the use of _________.

Accepted Answer

Variance Inflation Factors (VIFs) are used to measure the amount of multicollinearity in a set of multiple regression variables, making them a useful technique for controlling multicollinearity.

Question 19

Inspection of the following table of correlation coefficients for variables in a multiple regression analysis reveals potential multicollinearity with variables ___________. $$\begin{array} { | c | c | c | c | c | c | c | } 
\hline & y & x _ { 1 } & x _ { 2 } & x _ { 3 } & x _ { 4 } & x _ { 5 } \
\hline y & 1 & & & & & \
\hline x _ { 1 } & 0.854168 & 1 & & & & \
\hline x _ { 2 } & - 0.11828 & - 0.00383 & 1 & & & \
\hline x _ { 3 } & - 0.12003 & - 0.08499 & - 0.14523 & 1 & & \
\hline x _ { 4 } & 0.525901 & 0.118169 & - 0.14876 & 0.050042 & 1 & \
\hline x _ { 5 } & - 0.18105 & - 0.07371 & 0.995886 & - 0.14151 & - 0.16934 & 1 \
\hline
\end{array}$$

Accepted Answer

Variables $x_2$ and $x_5$ have a very high correlation coefficient of 0.995886, indicating potential multicollinearity between them.

Question 20

Inspection of the following table of correlation coefficients for variables in a multiple regression analysis reveals potential multicollinearity with variables ___________. $$\begin{array} { | c | r | c | c | c | c | c | } 
\hline & y & x _ { 1 } & x _ { 2 } & x _ { 3 } & x _ { 4 } & x _ { 5 } \
\hline y & 1 & & & & & \
\hline x _ { 1 } & - 0.08301 & 1 & & & & \
\hline x _ { 2 } & 0.236745 & - 0.51728 & 1 & & & \
\hline x _ { 3 } & 0.155149 & - 0.22264 & - 0.00734 & 1 & & \
\hline x _ { 4 } & 0.022234 & - 0.58079 & 0.884216 & 0.131956 & 1 & \
\hline x _ { 5 } & 0.4808 & - 0.20467 & 0.078916 & 0.207477 & 0.103831 & 1 \
\hline
\end{array}$$

Accepted Answer

The correlation coefficient between $x_2$ and $x_4$ is 0.884216, indicating a high degree of linear relationship, which suggests potential multicollinearity.

Quiz 14: Building Multiple Regression Models