Question 1

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.

Accepted Answer

The answer of The scatterplot of X and Y are...

Question 2

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

Accepted Answer

In multiple regression analysis, the best prediction is achieved when the independent variables have high correlations with the dependent variable and low correlations with each other. Choice C shows high correlations between the independent variables and the dependent variable (0.6 and 0.4) and a low correlation between the independent variables themselves (0.1), minimizing multicollinearity and enhancing the model's predictive power.

Question 3

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 = b₁X₁_i + b₂X₂_i + b₃X₃_i + 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.

Accepted Answer

The assumption of noncollinearity is violated because the model includes both individual quiz scores and their average, which are highly correlated variables. This can lead to multicollinearity issues in the regression analysis.

Question 4

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).

Accepted Answer

An interaction between two variables (in this case, X₁ and X₂) implies that the effect of one variable on the outcome (GPA, Y) depends on the level of the other variable. Choice B describes a situation where the relationship between GPA and time spent on watching TV changes based on SAT scores, indicating an interaction.

Question 5

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

Accepted Answer

When using categorical variables in a regression model, one less than the total number of categories needs to be dummy coded to avoid multicollinearity. Since there are six categories, only five need to be dummy coded.

Question 6

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.

Accepted Answer

The answer of You are given the following data, where...

Question 7

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.

Accepted Answer

The answer of Calculate the partial correlation r_12.3 and the...

Question 8

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:
 $\begin{array}{lllllll}
\hline \text { Source } & S S & d f & M S & F & \text { Critical Value } & \text { Decision } \
\hline \begin{array}{c}
\text { Regression } \
\text { Residual } \
\text { Total }
\end{array} & & & 6.5 & & & \
\hline
\end{array}$

Accepted Answer

The answer of A researcher would like to predict GPA...

Question 9

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.

Accepted Answer

The answer of You are given the following data, where...

Quiz 18: Multiple Regression

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

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

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

Calculate the partial correlation r12.3 and the part correlation r1(2.3) from the following bivariate correlations: r12 = .3, r13 = -.5, r23 = -.8.

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

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.