Question 1

For a multiple regression model:
A) SSY = SSR - SSE.
B) SSE = SSR - SSY.
C) SSR = SSE - SSY.
D) SSY = SSE + SSR.

Accepted Answer

In a multiple regression model, the total sum of squares (SSY) is equal to the sum of the regression sum of squares (SSR) and the error sum of squares (SSE). Therefore, SSY = SSR + SSE, making D the correct choice.

Question 2

In a multiple regression model, the probability distribution of the error variable $\varepsilon$ is assumed to be:
A) normal.
B) non-normal.
C) positively skewed.
D) negatively skewed.

Accepted Answer

The normality assumption of the error variable is a key assumption of multiple regression. If the error variable is not normally distributed, it may affect the validity of the regression results. Therefore, it is best to assume that the error variable is normally distributed in a multiple regression model.

Question 3

Which of the following best describes the range of the coefficient of multiple determination?
A) 0 to 1
B) −1 to 1
C) 1.0 to n, where n is the number of observations in the sample.
D) 1 to k, where k is the number of independent variables in the model.

Accepted Answer

The coefficient of multiple determination (R-squared) ranges from 0 to 1, where 0 indicates no linear relationship between the independent variable(s) and the dependent variable and 1 indicates a perfect linear relationship.

Question 4

In a multiple regression model, the following statistics are given: SSE = 100, $R ^ { 2 } = 0.995$ , k = 5, n = 15.
The multiple coefficient of determination adjusted for degrees of freedom is:
A) 0.955.
B) 0.992.
C) 0.930.
D) None of these choices are correct.

Accepted Answer

The formula for the adjusted coefficient of determination is:

$$R_{adj}^2 = 1 - \frac{SSE/(n-k-1)}{SST/(n-1)}$$

where SSE is the sum of squares due to error, SST is the total sum of squares, k is the number of independent variables, and n is the sample size.

Substituting the given values into the formula, we get:

$$R_{adj}^2 = 1 - \frac{100/(15-5-1)}{SST/(15-1)} = 0.992$$

Therefore, the best choice is B.

Question 5

A multiple regression model has the form ŷ = b₀+ b₁x₁ + b₂x₂. Which of the following best describes b₂? A) We estimate for each one unit increase in x₂, that y will increase by b₂ units, on average. B) Whilst holding x₁constant, we estimate for each one unit increase in x₂, that y will increase by b₂ units, on average. C) Whilst holding x₂constant, we estimate for each one unit increase in x₁, that y will increase by b₂ units, on average. D) Whilst holding x₁constant, we estimate for each one unit increase in x₂, that y will increase by b₁ units, on average.

Accepted Answer

b₂ represents the change in the dependent variable y for a one-unit change in x₂, while holding x₁ constant.

Question 6

In a multiple regression analysis involving 40 observations and 5 independent variables, total variation in y = SSY = 350 and SSE = 50. The multiple coefficient of determination is:
A) 0.8408.
B) 0.8571.
C) 0.8469.
D) 0.8529.

Accepted Answer

The multiple coefficient of determination is the ratio of the explained variance to the total variance:

R^2 = (SSY - SSE) / SSY

Plugging in the values, we get:

R^2 = (350 - 50) / 350 = 0.8571

Therefore, the best choice is B, with an R^2 of 0.8571.

Question 7

If multicollinearity exists among the independent variables included in a multiple regression model, then:
A) regression coefficients will be difficult to interpret.
B) the standard errors of the regression coefficients for the correlated independent variables will increase.
C) multiple coefficient of determination will assume a value close to zero.
D) the standard errors of the regression coefficients for the correlated independent variables will increase and and multiple coefficient of determination will assume a value close to zero.

Accepted Answer

Multicollinearity makes it difficult to interpret the regression coefficients and leads to an increase in the standard errors of the regression coefficients for the correlated independent variables. Additionally, multicollinearity reduces the ability of the model to explain the variation in the dependent variable, leading to a lower multiple coefficient of determination. Therefore, option D is the correct answer.

Question 8

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) 5 and 25.
B) 24 and 5.
C) 25 and 5.
D) 5 and 24.

Accepted Answer

The answer of In testing the validity of a multiple...

Question 9

In a multiple regression model, the error variable $ { \varepsilon }$ 
 is assumed to have a mean of:
A) -1.0.
B) 0.0.
C) 1.0.
D) any value smaller than -1.0.

Accepted Answer

The error variable, also known as the residual, is the difference between the predicted and actual values of the dependent variable. In a multiple regression model, the goal is to minimize the sum of squared residuals. Since the residuals can be positive or negative, they will cancel each other out when aggregated. Therefore, the mean of the residuals is expected to be close to zero if the model is well-specified and the errors are normally distributed.

Question 10

The graphical depiction of the equation of a multiple regression model with k independent variables (k > 1) is referred to as:
A) a straight line.
B) the response variable.
C) the response surface.
D) a plane only when k = 3.

Accepted Answer

The graphical depiction of the equation of a multiple regression model with more than one independent variable (k > 1) is called a response surface, which is a three-dimensional plot with the response variable on the z-axis and the independent variables on the x and y axes. It is not a straight line or the response variable itself, and it can be created for any number of independent variables, not just when k = 3. Therefore, the best choice is C.

Question 11

Which of the following measures can be used to assess a multiple regression model's fit?
A) The sum of squares for error.
B) The sum of squares for regression.
C) The standard error of estimate.
D) A single t-test.

Accepted Answer

The answer of Which of the following measures can be...

Question 12

If none of the data points for a multiple regression model with two independent variables were on the regression plane, then the multiple coefficient of determination would be:
A) -1.0.
B) 1.0.
C) any number between -1 and 1, inclusive.
D) any number greater than or equal to zero but smaller than 1.

Accepted Answer

The multiple coefficient of determination (R-squared) measures the proportion of variation in the dependent variable that is explained by the independent variables in the model. If none of the data points lie on the regression plane, it means that the model cannot explain any of the variation in the dependent variable. Therefore, the R-squared value would be 0, which is a number greater than or equal to zero but smaller than 1. Hence, the correct answer is D.

Question 13

Which of the following best describes a multiple linear regression model?
A) A multiple linear regression model has more than one dependent variable.
B) A multiple linear regression model has more than one independent variable.
C) A multiple linear regression model must have more than one independent variable and more than one independent variable.
D) A multiple linear regression model has one independent variable.

Accepted Answer

A multiple linear regression model is a statistical model that uses more than one independent variable to predict a dependent variable. Therefore, option B best describes a multiple linear regression model. Option A is incorrect as the dependent variable in a multiple linear regression model is only one. Option C is incorrect as it includes the incorrect statement that a model must have more than one dependent variable. Option D is incorrect as it states that a multiple linear regression model has only one independent variable, which is not true.

Question 14

An estimated multiple regression model has the form ŷ = 100 − 2x₁+ 9x₂. As x₁ increases by 1 unit while holding x₂ constant, which of the following best describes the change in y? A) y will increase by 2 units, estimated, on average. B) y will decrease by 98 units, estimated, on average. C) y will increase by 98 units, estimated, on average. D) y will decrease by 2 units, estimated, on average.

Accepted Answer

As x₁ increases by 1 unit, the term -2x₁ causes y to decrease by 2 units, estimated, on average, while holding x₂ constant.

Question 15

For a multiple regression model with n = 35 and k = 4, the following statistics are given: SS_y = 500 and SSE = 100. The coefficient of determination is: A) 0.82. B) 0.80. C) 0.77. D) 0.20.

Accepted Answer

The answer of For a multiple regression model with n...

Question 16

In a multiple regression analysis involving 20 observations and 5 independent variables, total variation in y = SSY = 250 and SSE = 35. The multiple coefficient of determination, adjusted for degrees of freedom, is:
A) 0.810.
B) 0.860.
C) 0.835.
D) 0.831.

Accepted Answer

The answer of In a multiple regression analysis involving 20...

Question 17

For a multiple regression model, the following statistics are given: Total variation in y = SSY = 250, SSE = 50, k = 4, n = 20.
The coefficient of determination adjusted for degrees of freedom is:
A) 0.800.
B) 0.747.
C) 0.840.
D) 0.775.

Accepted Answer

The coefficient of determination adjusted for degrees of freedom is given by:

R^2adj = 1 - (SSE/(n-k-1)) / (SSY/(n-1))

Plugging in the values given, we get:

R^2adj = 1 - (50/(20-4-1)) / (250/(20-1))
R^2adj = 0.747

Therefore, the answer is B.

Question 18

Which of the following is not true when we add an independent variable to a multiple regression model?
A) The adjusted coefficient of determination can assume a negative value.
B) The unadjusted coefficient of determination always increases.
C) The unadjusted coefficient of determination may increase or decrease.
D) The adjusted coefficient of determination may increase.

Accepted Answer

When an independent variable is added to a multiple regression model, the unadjusted coefficient of determination (R-squared) always increases or stays the same because it reflects the proportion of the variance in the dependent variable that is predictable from the independent variables. It does not decrease because adding a variable can only explain more variance, not less, even if the new variable is only marginally useful.

Question 19

The multiple coefficient of determination is defined as:
A) SSE/SSY.
B) MSE/MSR.
C) 1 - (SSE/SSY).
D) 1 - (MSE/MSR).

Accepted Answer

The multiple coefficient of determination, often denoted as R², is calculated as 1 minus the ratio of the sum of squares for error (SSE) to the total sum of squares (SSY). This represents the proportion of variance in the dependent variable that is predictable from the independent variables.

Question 20

For the estimated multiple regression model = 30 -4x₁+ 5x₂+3 x₃, a one unit increase in x₃, holding x₁ and x₂ constant, will result in which of the following changes in y? A) y will increase by 3 units. B) y will increase by 2 units, estimated, on average. C) y will increase by 33 units D) y will increase by 3 units, estimated, on average.

Accepted Answer

According to the estimated regression equation, a one unit increase in x₃ will result in a coefficient of 3 on y, holding x₁ and x₂ constant. Therefore, y will increase by 3 units, estimated, on average, when x₃ increases by one unit.

Quiz 19: Multiple Regression

For a multiple regression model:

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

Which of the following best describes the range of the coefficient of multiple determination?

In a multiple regression model, the following statistics are given: SSE = 100, $R ^ { 2 } = 0.995$ , k = 5, n = 15. The multiple coefficient of determination adjusted for degrees of freedom is:

A multiple regression model has the form ŷ = b₀+ b₁x₁ + b₂x₂. Which of the following best describes b₂?

In a multiple regression analysis involving 40 observations and 5 independent variables, total variation in y = SSY = 350 and SSE = 50. The multiple coefficient of determination is:

If multicollinearity exists among the independent variables included in a multiple regression model, then:

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:

In a multiple regression model, the error variable ${ \varepsilon }$ is assumed to have a mean of:

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 measures can be used to assess a multiple regression model's fit?

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

Which of the following best describes a multiple linear regression model?

An estimated multiple regression model has the form ŷ = 100 − 2x₁+ 9x₂. As x₁ increases by 1 unit while holding x₂ constant, which of the following best describes the change in y?

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

In a multiple regression analysis involving 20 observations and 5 independent variables, total variation in y = SSY = 250 and SSE = 35. The multiple coefficient of determination, adjusted for degrees of freedom, is:

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

Which of the following is not true when we add an independent variable to a multiple regression model?

The multiple coefficient of determination is defined as:

For the estimated multiple regression model = 30 -4x₁+ 5x₂+3 x₃, a one unit increase in x₃, holding x₁ and x₂ constant, will result in which of the following changes in y?