Deck 18: Multiple Regression

Question 1

Variable 1 is to be predicted from a combination of variable 2 and one of variables 3, 4, 5, and 6. The correlations of importance are as follows:
R₁₃ = .3; r₂₃ = .9
R₁₄ = .4; r₂₄ = .2
R₁₅ = .6; r₂₅ = .8
R₁₆ = .7; r₂₆ = .1
Which of the following multiple correlation coefficients will have the smallest value?

A) r_1.23
B) r_1.24
C) r_1.25
D) r_1.26

Answer 1

A

Question 2

Carol is studying the correlation of college GPA (X₁) and the number of hours spent on watching TV (X₂). As intelligence is expected to affect GPA, she would like to remove the influence of IQ (X₃) from GPA scores when computing the correlation. This is an example of which one of the following?

A) Bivariate correlation
B) Partial correlation
C) Semipartial correlation
D) Regression correlation

Answer 2

C

Question 3

The correlation of GPA (X₁) and the number of hours spent on watching TV (X₂) where the influence of IQ (X₃) is removed from GPA only can be denoted by

A) r_2.13
B) r_12.3
C) r_13.2
D) r_1(2.3)
E) r_2(1.3)

Answer 3

E

Question 4

Suppose for the three variables, GPA (X₁), time spent on watching TV (X₂), and IQ (X₃), the bivariate correlation coefficients are computed as below:
R₁₂ = -0.4; r₁₃ = 0.6; r₂₃ = 0.
If we remove the influence of IQ from GPA, the correlation of GPA and the time spent on watching TV will be

A) stronger than the bivariate correlation r₁₂.
B) weaker than the bivariate correlation r₁₂.
C) the same as the bivariate correlation r₁₂.
D) uncertain.

Question 5

David is studying the correlation of temperature (X₁) and the consumption of ice cream (X₂). As he expects that the price of ice cream (X₃) is correlated with both temperature and the consumption of ice cream, he would like to remove the effects of price from both X₁ and X₂ when computing the correlation. This is an example of which one of the following?

A) Bivariate correlation
B) Partial correlation
C) Semipartial correlation
D) Regression correlation

Question 6

The correlation between temperature (X₁) and the consumption of ice cream (X₂) controlling for price of ice cream (X₃) can be denoted by

A) r_2.13
B) r_12.3
C) r_13.2
D) r_1(2.3)
E) r_2(1.3)

Question 7

David states that the correlation of the price of ice cream and the consumption of ice cream is -0.3 when temperature is held constant. By saying "held constant", David is implying that

A) he conducted an experiment in a laboratory where the temperature can be artificially controlled.
B) the correlation of price and consumption of ice cream is -0.3 when the effects of temperature are removed.
C) the conclusion is reached using data collected at the same temperature.
D) the correlation of price and consumption of ice cream is stronger when the temperature stays the same than when the temperature varies.

Question 8

For three variables, X₁, X₂, and X₃, the bivariate correlation coefficients are:
R₁₂ = 0.7; r₁₃ = 0; r₂₃ = 0.
The correlation of X₁ and X₂ controlling for X3 will be

A) stronger than the bivariate correlation r₁₂.
B) weaker than the bivariate correlation r₁₂.
C) the same as the bivariate correlation r₁₂.
D) uncertain.

Question 9

The regression line for predicting selling price of houses (in $1000) (Y) from size of the house (in 1000 square feet) (X₁) and number of bathrooms (X₂) is found to be Y = -41.8 + 64.8X₁ + 19.2X₂ + e_i. Which of the following statements is a correct interpretation of the equation?

A) Compared to a house with only one bathroom, a house with two bathrooms will be $19200 higher in terms of selling price.
B) For two houses of the same size, the house with one additional bathroom is expected to be $19200 higher in its selling price.
C) Larger houses are expected to have higher selling prices than smaller houses, regardless of the number of bathrooms.
D) To increase the selling price of his house, the home owner should build as many bathrooms as he/she can in the house.

Question 10

In the scenario described in Question 9, if the residual (e_i) is -3.5 for a particular house, it means that

A) the regression function overestimates the selling price by $3,500.
B) the regression function underestimates the selling price by $3,500.
C) the selling price is estimated to be 3.5 standard deviations below the average price.
D) the regression function is not valid because the prediction is erroneous.

Question 11

The multiple regression model for predicting Y from X₁, X₂, and X₃ is
Y_i = b₁X_1i + b₂X_2i + b₃X_3i + a + e_i. If the bivariate correlation of Y and X₁ is positive (r_Y1 > 0), the partial slope for X₁ (b₁) will be

A) a positive value.
B) a negative value.
C) zero.
D) uncertain.

Question 12

In multiple regression, if the null hypothesis, H₀: $\beta$ ₁ = $\beta$ ₂ = $\beta$ ₃ = $\beta$ ₄ = 0, is rejected, it means that

A) there is no linear relationship between Y and any of the independent variables.
B) there is a linear relationship between Y and all four independent variables.
C) there is a linear relationship between Y and at least one independent variable.
D) none of the individual regression coefficients (b_k) are significantly different from 0.
E) all of the individual regression coefficients (b_k) are significantly different from each other.

Question 13

In a multiple regression model, Y is predicted from X₁, X₂, and X₃. Both R² and R_adj² are computed. If X₃ is removed from the model, how will R² change?

A) R² will increase.
B) R² will decrease.
C) R² will not change.
D) uncertain.

Question 14

In a multiple regression model, Y is predicted from X₁, X₂, and X₃. Both R² and R_adj² are computed. If X₃ is removed from the model, how will R_adj² change?

A) R_adj² will increase.
B) R_adj² will decrease.
C) R_adj² will not change.
D) uncertain.

Question 15

For the regression model, Y_i = b₁X_1i + b₂X_2i + a + e_i, consider the following two situations:
Situation 1: r_Y1 = -0.5 r_Y2 = 0.8 r₁₂ = 0.1
Situation 2: r_Y1 = -0.5 r_Y2 = 0.8 r₁₂ = 0.3
In which of the two situations will R² be larger?

A) situation 1.
B) situation 2.
C) R² will be the same in both situations.
D) uncertain.

Question 16

For the regression model, Y_i = b₁X_1i + b₂X_2i + a + e_i, consider the following two situations:
Situation 1: r_Y1 = -0.5 r_Y2 = 0.8 r₁₂ = 0.1
Situation 2: r_Y1 = 0.2 r_Y2 = 0.8 r₁₂ = 0.1
In which of the two situations will R² be larger?

A) situation 1.
B) situation 2.
C) R² will be the same in both situations.
D) uncertain.

Question 17

In a multiple regression, the F test for the overall model is highly significant, but none of the t values for individual predictors are significant. What is the most likely cause for this situation?

A) heterogeneous variances.
B) non-independence of residuals.
C) non-normality of residuals.
D) non-linearity relation between Y and the predictors.
E) collinearity of the independent variables.

Question 18

All of the following are possible effects of multicollinearity except

A) the standard error of the regression coefficients may be larger than expected.
B) the signs of the regression coefficients may be opposite of what is expected.
C) regression coefficients can be quite unstable across samples.
D) R² may be significant, yet none of the predictors are significant.
E) the VIF is 0 for all predictors.

Question 19

Which of the following statements about R² and R_adj² is correct?

A) R² is always larger than R_adj².
B) R² will always increase as more predictors are added to the model.
C) R_adj² adjusts for the number of independent variables and sample size.
D) If an additional independent variable were entered in the model, an increase in R² indicates the new variable is adding value to the model.
E) R² and R_adj² will never be negative.

Question 20

Carol is building a multiple regression model to predict college GPA from a set of predictors. There is theory suggesting that college GPA can be predicted by students' involvement in college when controlling for their prior achievement. Therefore, Carol first entered into the model a set of variables that measure prior achievement (e.g., high school GPA, SAT), and then added a set of variables that measure collegial involvement. Which one of the following procedures is used?

A) Backward elimination
B) Forward selection
C) Stepwise selection
D) Hierarchical regression

Question 21

The scatterplot of X and Y are shown as below.
Based on the plot, which model is the most appropriate to use?

A) Y_i = b₁X_i + a + e_i.
B) Y_i = b₁X_i + b₂X_i² + a + e_i.
C) Y_i = b₁X_i² + a + e_i.
D) Y_i = b₁X_i + b₂X_i + b₃X_i³ + a + e_i.

Question 22

Which of the following situations will result in the best prediction in multiple regression analysis?

A) r_Y1 = 0.1 r_Y2 = 0.4 r₁₂ = 0.1
B) r_Y1 = 0.1 r_Y2 = 0.4 r₁₂ = 0.8
C) r_Y1 = 0.6 r_Y2 = 0.4 r₁₂ = 0.1
D) r_Y1 = 0.6 r_Y2 = 0.4 r₁₂ = 0.8

Question 23

An instructor wanted to know if the scores on pop quizzes are good predictors of the scores on the final exam. He used the following regression model,
Y_i = b₁X_1i + b₂X_2i + b₃X_3i + a + e_i, where Y is the score on the final exam, X₁ is the score on the first quiz, X₂ is the score on the second quiz, and X₃ is the average score of the two quizzes. Evaluate this model.

A) The assumption of independence is violated.
B) The assumption of linearity is violated.
C) The assumption of noncollinearity is violated.
D) There is no indication of assumption violation based on the information given.

Question 24

An interaction between X₁ and X₂ is present in which of the following situations?

A) The relationship between GPA (Y) and the time spent on watching TV (X₁) is similar for students with different SAT scores (X₂).
B) The relationship between GPA (Y) and the time spent on watching TV (X₁) is different for students with higher SAT scores versus those with lower SAT scores (X₂).
C) There is a strong correlation between the time spent on watching TV (X₁) and SAT scores (X₂).
D) There is a strong correlation between GPA (Y) and the time spent on watching TV (X₁) controlling for SAT scores (X₂).

Question 25

Karen wants to use a categorical variable, levels of education, to predict annual income. There are six categories in the levels of education: High school graduate, Some college, Associate's degree, Bachelor's degree, Master's degree, Doctorate, or professional degree. How many categories need to be dummy coded and included in the regression model as predictors?

A) 4
B) 5
C) 6
D) 7

Question 26

You are given the following data, where X₁ (Pretest score) and X₂ (Hours spent in the program) are used to predict Y (Posttest score):
$\begin{array}{ccc}\hline \boldsymbol{Y} & \boldsymbol{X}_{\mathbf{1}} & \boldsymbol{X}_{\mathbf{2}} \\\hline 65 & 60 & 7.5 \\82 & 62 & 9.0 \\94 & 75 & 8.5 \\80 & 78 & 7.0 \\87 & 65 & 10.0 \\66 & 60 & 8.0 \\\hline\end{array}$ Determine the following values: intercept, b₁, b₂, SS_res, SS_reg, F, s_res², s(b₁), s(b₂), t₁, t₂.

Question 27

Calculate the partial correlation r_12.3 and the part correlation r_1(2.3) from the following bivariate correlations: r₁₂ = .3, r₁₃ = -.5, r23 = -.8.

Question 28

A researcher would like to predict GPA from a set of three predictor variables for a sample of 34 college students. Multiple linear regression analysis was utilized. Complete the following summary table ( $\alpha$ = .05) for the test of significance of the overall regression model:
$\begin{array}{lllllll}\hline \text { Source } & S S & d f & M S & F & \text { Critical Value } & \text { Decision } \\\hline \begin{array}{c}\text { Regression } \\\text { Residual } \\\text { Total }\end{array} & & & 6.5 & & & \\\hline\end{array}$

Question 29

You are given the following data, where X₁ (attendance rate) and X₂ (average SAT score) are to be used to predict Y (average score in graduation test). Each case represents one school.
$\begin{array}{|c|c|c|}\hline Y & X_{\mathbf{1}} & X_{\mathbf{2}} \\\hline 78.4 & 93.4 & 1010 \\\hline 81.3 & 94.6 & 1020 \\\hline 81.3 & 95.4 & 1024 \\\hline 82.5 & 91.1 & 1136 \\\hline 77.8 & 91.6 & 952 \\\hline 84.5 & 94.2 & 1042 \\\hline 88.2 & 94.5 & 1106 \\\hline 88.7 & 93.4 & 1004 \\\hline 72.5 & 92.1 & 880 \\\hline 85.4 & 94.9 & 1124 \\\hline 82.9 & 94.3 & 1124 \\\hline 81.4 & 94.7 & 996 \\\hline\end{array}$ Determine the following values: intercept, b₁, b₂, SS_res, SS_reg, F, s_res², s(b₁), s(b₂), t₁, t₂.