Question 1

In reference to the equation: $\hat{ Y }$ = -0.25 + 0.08x₁ + 0.10x₂,the value 0.08 is the: A) predicted value of y. B) estimated change in y when x₁ increases by one unit. C) estimated change in y when x₂ increases by one unit. D) predicted value of y when x₁ = 0 and x₂ = 0. E) predicted value of y when x₁= 2 and x₂ = 1.

Accepted Answer

In linear regression, the coefficients of predictor variables estimate the change in the response variable for a one unit increase in the predictor variable, holding all other variables constant. Therefore, the value 0.08 represents the estimated change in y when x₁ increases by one unit.

Question 2

The adjusted multiple coefficient of determination is adjusted for the:
A) number of regression parameters including the y-intercept.
B) number of dependent variables and the sample size.
C) number of independent variables and the sample size.
D) coefficient of correlation and the significance level.

Accepted Answer

The adjusted multiple coefficient of determination, also known as the adjusted R-squared, is adjusted for the number of independent variables and the sample size. It takes into account the possibility of overfitting, which can happen when adding too many independent variables to a model. Therefore, it is only adjusted for the independent variables, not the y-intercept or the dependent variables. The coefficient of correlation and significance level are not specifically adjusted for by the adjusted R-squared.

Question 3

A multiple regression equation includes 3 independent variables,and the coefficient of multiple determination is 0.64.The percentage of the variation in y that is explained by the regression equation is:
A) 64%.
B) 80%.
C) 36%.
D) 21%.

Accepted Answer

The coefficient of multiple determination, denoted by R-squared, represents the proportion of variance in the dependent variable (y) that is explained by the independent variables in the regression equation. 
In this case, R-squared is 0.64 or 64%. Therefore, the regression equation explains 64% of the variation in the dependent variable.

Question 4

For the multiple regression model $\hat { Y }$ = 50 + 25x₁ - 10x₂ + 8x₃,if x₂ were to increase by 5,holding x₁ and x₃ constant,the value of y would: A) increase by 5. B) increase by 50. C) decrease on average by 5. D) decrease on average by 50.

Accepted Answer

According to the regression model, an increase in x₂ leads to a decrease in the predicted value of $\hat { Y }$ by 10. Therefore, if x₂ were to increase by 5, the predicted value of $\hat { Y }$ would decrease by (10*5) = 50.

Question 5

The purpose of the multiple correlation analysis is to measure the strength of the relationship between the dependent (y)and the set of independent (x)variables.

Accepted Answer

This is the main purpose of multiple correlation analysis.

Question 6

A multiple regression analysis includes 25 data points and 4 independent variables results in SST = 200 and SSR = 150.The multiple standard error of estimate will be:
A) 1.333.
B) 6.124.
C) 2.500.
D) 1.581.

Accepted Answer

The multiple standard error of estimate is calculated using the formula: sqrt(SSE / (n - k - 1)), where SSE = SST - SSR, n is the number of observations, and k is the number of independent variables. Here, SSE = 200 - 150 = 50, n = 25, and k = 4. Thus, the multiple standard error of estimate = sqrt(50 / (25 - 4 - 1)) = sqrt(50 / 20) = sqrt(2.5) = 1.581.

Question 7

A dummy variable is used to incorporate qualitative data into the analysis.

Accepted Answer

A dummy variable, also known as an indicator variable, is a binary variable that takes the value 0 or 1 to represent the absence or presence of a particular characteristic. It is commonly used to incorporate categorical data into regression models or other statistical analyses.

Question 8

Multicollinearity is a situation in which two or more of the independent variables are highly correlated with each other.

Accepted Answer

Multicollinearity occurs when there is a high correlation between two or more independent variables in a regression model. This can create problems with interpreting the coefficients and can lead to unstable and inaccurate predictions.

Question 9

In a multiple regression model,the following statistics are given: SSE = 100,R² = 0.995,k = 5,n = 15.The multiple coefficient of determination adjusted for degrees of freedom is: A) 0.955. B) 0.992. C) 0.900. D) 0.855.

Accepted Answer

The formula for the adjusted coefficient of determination is: $$R_{adj}^2 = 1 - \frac{(n-1)}{(n-k)}\frac{SSE}{SST}$$ where SSE is the sum of squared errors, SST is the total sum of squares, k is the number of predictors in the model, and n is the sample size. We are given that SSE = 100, k = 5, and n = 15. We need to find SST to calculate $R_{adj}^2$. The total sum of squares can be calculated as: $$SST = \sum_{i=1}^{n}(y_i - \bar{y})^2$$ where $y_i$ is the observed value of the dependent variable and $\bar{y}$ is the mean of the dependent variable. Since we don't have any information about the dependent variable, we need to assume that the model includes an intercept term. In this case, we can use the fact that SSE is equal to the sum of squared errors around the mean: $$SSE = \sum_{i=1}^{n}(y_i - \bar{y})^2$$ Therefore, we can calculate $\bar{y}$ as: $$\bar{y} = \frac{1}{n}\sum_{i=1}^{n}y_i$$ We don't have any information about the values of the independent variables, so we can't calculate the predicted values of the dependent variable or the residuals. However, we know that the multiple coefficient of determination is: $$R^2 = \frac{SSR}{SST}$$ where SSR is the sum of squared regression (also known as explained sum of squares). Since we don't have the values of the independent variables, we can't calculate SSR directly. However, we are given that: $$R² = R = \sqrt{R^2} = 0.995$$ This means that the model explains 99.5% of the variability in the dependent variable. Therefore, we can estimate SST as: $$SST = \frac{SSE}{1 - R^2} = \frac{100}{1 - 0.995^2} = 5025$$ Plugging in the values into the formula for $R_{adj}^2$, we get: $$R_{adj}^2 = 1 - \frac{(n-1)}{(n-k)}\frac{SSE}{SST} = 1 - \frac{14}{10}\frac{100}{5025} = 0.992$$ Therefore, the adjusted coefficient of determination is 0.992, which is closest to choice B).

Question 10

It is a good idea to make point estimates based on x values that lie beyond the range of the underlying data.

Accepted Answer

The answer of It is a good idea to make...

Question 11

In reference to the equation: $\hat { Y }$ = -0.25 + 0.08x₁ + 0.10x₂,the value -0.25 is the: A) predicted value of y. B) partial regression coefficient for x₁. C) partial regression coefficient for x₂. D) predicted value of y when x₁ = 0 and x₂ = 0. E) predicted value of when x₁ = 2 and x₂ = 1.

Accepted Answer

The value -0.25 is the intercept of the regression equation, representing the predicted value of $\hat { Y }$ when all independent variables (x₁ and x₂) are equal to 0.

Question 12

In reference to the equation: $\hat { y }$ = -0.25 + 0.08x₁ + 0.10x₂,the value 0.01 is the: A) predicted value of y. B) partial regression coefficient for x_1. C) partial regression coefficient for x_2. D) predicted value y when x₁ = 0 and x₂ = 0. E) predicted value of y when x₁ = 2 and x₂ = 1.

Accepted Answer

The value 0.01 does not explicitly appear in the given equation, but the correct interpretation of the equation's components and their relationship to the variables and constants can lead to the calculation of specific predicted values of y under given conditions.

Question 13

In a multiple regression analysis involving 40 observations and 4 independent variables,SST = 375 and SSE = 75.The multiple coefficient of determination is:
A) 0.8333.
B) 0.8000.
C) 0.1875.
D) 0.9375.

Accepted Answer

The answer of In a multiple regression analysis involving 40...

Question 14

A multiple regression model has the form: $\hat { Y }$ = 5.25 + 2.5x₁ + 4x₂.As x₂ increases by 1 unit,holding x₁ constant,then the value of y will increase by: A) 2.5 units. B) 7.75 units. C) 4 units on average. D) 11.75 units on average.

Accepted Answer

The coefficient of $x_{2}$ in the regression equation is 4, indicating that for each one-unit increase in $x_{2}$, holding $x_{1}$ constant, the predicted value of $y$ ($\hat{Y}$) increases by 4 units on average.

Question 15

In interpreting the multiple regression equation,it can be a mistake to conclude that one independent variable is more important than another just because its partial regression coefficient happens to be large.

Accepted Answer

A large partial regression coefficient indicates a strong relationship between that independent variable and the dependent variable, but it does not necessarily mean that it is more important than other independent variables. The importance of each independent variable should be evaluated based on a combination of statistical significance, theoretical relevance, and practical implications.

Question 16

A multiple regression model has the form $\hat { Y }$ = b₀ + b₁x₁ + b₂x₂.The coefficient b₁ is interpreted as the: A) estimated change in y per unit change in x₁. B) estimated change in y per unit change in x₁,holding x₂constant. C) estimated change in y per unit change in x₁,when x₁ and x₂ values are correlated. D) estimated change in y per unit change in x₂,holding x₁ constant.

Accepted Answer

The coefficient b₁ is the estimated change in y per unit change in x₁, holding x₂ constant. This is because b₁ is the coefficient for the x₁ term in the model, and multiple regression coefficients represent the change in y associated with a one-unit increase in the corresponding predictor variable, holding all other predictor variables constant.

Question 17

Multiple regression analysis examines the linear relationship between a dependent variable (y)and two or more independent variables (x₁,x₂,and so on).

Accepted Answer

Multiple regression analysis does involve examining the linear relationship between a dependent variable and two or more independent variables.

Question 18

For each y term in the multiple regression equation,the corresponding $\beta$ is referred to as the partial regression coefficient.

Accepted Answer

In a multiple regression equation, the corresponding $\beta$ for each x term (independent variable) is referred to as the partial regression coefficient, not the y term (dependent variable).

Question 19

The amount of variation in the dependent variable that is not explained by the multiple regression equation is known as:
A) total sum of squares.
B) residual sum of squares.
C) regression sum of squares.
D) treatment sum of squares.

Accepted Answer

The residual sum of squares (RSS) is the amount of variation in the dependent variable that is not explained by the regression equation. It represents the error or noise in the data that cannot be accounted for by the independent variables. This is also known as the sum of squared residuals or the sum of squared errors.

Question 20

For the multiple regression model $\hat{ Y }$ = 3 - 4x₁ + 5x₂ + 2x₃,a unit increase in x₁,holding x₂ and x₃ constant,results in: A) an increase of 4 units in the value of y. B) a decrease of 4 units in the value of y. C) a decrease of 4 units,on average,in the value of y. D) an increase of 6 units in the value of y.

Accepted Answer

The answer of For the multiple regression model \(\hat{ Y...

Quiz 16: Multiple Regression and Correlation

In reference to the equation: Y^\hat{ Y }Y^ = -0.25 + 0.08x1 + 0.10x2,the value 0.08 is the:

The adjusted multiple coefficient of determination is adjusted for the:

A multiple regression equation includes 3 independent variables,and the coefficient of multiple determination is 0.64.The percentage of the variation in y that is explained by the regression equation is:

For the multiple regression model Y^\hat { Y }Y^ = 50 + 25x1 - 10x2 + 8x3,if x2 were to increase by 5,holding x1 and x3 constant,the value of y would:

The purpose of the multiple correlation analysis is to measure the strength of the relationship between the dependent (y)and the set of independent (x)variables.

A multiple regression analysis includes 25 data points and 4 independent variables results in SST = 200 and SSR = 150.The multiple standard error of estimate will be:

A dummy variable is used to incorporate qualitative data into the analysis.

Multicollinearity is a situation in which two or more of the independent variables are highly correlated with each other.

In a multiple regression model,the following statistics are given: SSE = 100,R2 = 0.995,k = 5,n = 15.The multiple coefficient of determination adjusted for degrees of freedom is:

It is a good idea to make point estimates based on x values that lie beyond the range of the underlying data.

In reference to the equation: Y^\hat { Y }Y^ = -0.25 + 0.08x1 + 0.10x2,the value -0.25 is the:

In reference to the equation: y^\hat { y }y^​ = -0.25 + 0.08x1 + 0.10x2,the value 0.01 is the:

In a multiple regression analysis involving 40 observations and 4 independent variables,SST = 375 and SSE = 75.The multiple coefficient of determination is:

A multiple regression model has the form: Y^\hat { Y }Y^ = 5.25 + 2.5x1 + 4x2.As x2 increases by 1 unit,holding x1 constant,then the value of y will increase by:

In interpreting the multiple regression equation,it can be a mistake to conclude that one independent variable is more important than another just because its partial regression coefficient happens to be large.

A multiple regression model has the form Y^\hat { Y }Y^ = b0 + b1x1 + b2x2.The coefficient b1 is interpreted as the:

Multiple regression analysis examines the linear relationship between a dependent variable (y)and two or more independent variables (x1,x2,and so on).

For each y term in the multiple regression equation,the corresponding β\betaβ is referred to as the partial regression coefficient.

The amount of variation in the dependent variable that is not explained by the multiple regression equation is known as:

For the multiple regression model Y^\hat{ Y }Y^ = 3 - 4x1 + 5x2 + 2x3,a unit increase in x1,holding x2 and x3 constant,results in:

In reference to the equation: $\hat{ Y }$ = -0.25 + 0.08x₁ + 0.10x₂,the value 0.08 is the:

For the multiple regression model $\hat { Y }$ = 50 + 25x₁ - 10x₂ + 8x₃,if x₂ were to increase by 5,holding x₁ and x₃ constant,the value of y would:

In a multiple regression model,the following statistics are given: SSE = 100,R² = 0.995,k = 5,n = 15.The multiple coefficient of determination adjusted for degrees of freedom is:

In reference to the equation: $\hat { Y }$ = -0.25 + 0.08x₁ + 0.10x₂,the value -0.25 is the:

In reference to the equation: $\hat { y }$ = -0.25 + 0.08x₁ + 0.10x₂,the value 0.01 is the:

A multiple regression model has the form: $\hat { Y }$ = 5.25 + 2.5x₁ + 4x₂.As x₂ increases by 1 unit,holding x₁ constant,then the value of y will increase by:

A multiple regression model has the form $\hat { Y }$ = b₀ + b₁x₁ + b₂x₂.The coefficient b₁ is interpreted as the:

Multiple regression analysis examines the linear relationship between a dependent variable (y)and two or more independent variables (x₁,x₂,and so on).

For each y term in the multiple regression equation,the corresponding $\beta$ is referred to as the partial regression coefficient.

For the multiple regression model $\hat{ Y }$ = 3 - 4x₁ + 5x₂ + 2x₃,a unit increase in x₁,holding x₂ and x₃ constant,results in: