Question 1

A researcher who is evaluating the significance of a multiple regression equation with two predictor variables obtains an F-ratio with df = 2,24.How many individuals were needed to produce the sample data used to find the regression equation?​

Accepted Answer

The degrees of freedom for the F-ratio in a regression with two predictors is df = (k, n-k-1), where k is the number of predictors and n is the sample size. Here, df = (2, 24), so n - 2 - 1 = 24, which means n = 27.

Question 2

For a linear regression calculated for a sample of n = 20 pairs of X and Y values,what is the value for degrees of freedom for the predicted portion of the Y-score variance,MS_regression?

Accepted Answer

The degrees of freedom for the predicted portion of the Y-score variance in a simple linear regression is always 1, because it is calculated based on the regression line itself, which is defined by two parameters (the slope and the intercept).

Question 3

A researcher who is evaluating the significance of a multiple regression equation with two predictor variables obtains an F-ratio with df = 2,18.How many individuals were needed to produce the sample data used to find the regression equation?​

Accepted Answer

The degrees of freedom for the regression (numerator) is equal to the number of predictor variables, which is 2. The degrees of freedom for the error (denominator) is equal to the total number of observations minus the number of parameters estimated (including the intercept), which is 18. Therefore, the total number of observations is 18 + 2 + 1 = 21.

Question 4

A multiple regression equation with two predictor variables produces R² = 0.10.What portion of the variability for the Y scores is predicted by the equation?

Accepted Answer

The coefficient of determination, denoted as R^2, represents the proportion of the variance for a dependent variable that's predictable from the independent variables. In this case, R² = 0.10 or R^2 = 0.10, which means 10% of the variability in Y scores is predicted by the equation.

Question 5

The Pearson correlation between X₁ and Y is r = 0.50 and the correlation between X₂ and Y is r = 0.20.How much variance for the Y scores is predicted by using both X₁ and X₂ as predictor variables in a multiple regression equation?

Accepted Answer

The information provided does not include the correlation between X₁ and X₂, which is necessary to calculate the combined variance explained in Y by both predictors in a multiple regression equation.

Question 6

For a linear regression calculated for a sample of n = 20 pairs of X and Y values,what is the value for degrees of freedom for the unpredicted portion of the Y-score variance,MS_residual?

Accepted Answer

The degrees of freedom for the unpredicted portion of the Y-score variance in linear regression is equal to the sample size minus the number of estimated regression coefficients (including the intercept), which is n - p. In this case, p = 2 (one for the intercept and one for the slope), so the degrees of freedom for the unpredicted portion of the Y-score variance is 20 - 2 = 18. Therefore, the correct answer is C.

Question 7

The distance between the Y value in the data and the Y value predicted from the regression equation is known as the residual.What is the value for the sum of the squared residuals?​

Accepted Answer

The sum of the squared residuals is given by the formula $SS_{residual} = (1 - r^2)(SS_Y)$, where $r^2$ is the coefficient of determination and $SS_Y$ is the total sum of squares for the dependent variable. This formula reflects the proportion of the variance in the dependent variable that is not explained by the model.

Question 8

For either linear regression or multiple regression,the standard error of estimate can be computed as ____.​

Accepted Answer

The standard error of estimate for either linear regression or multiple regression is computed as the square root of the mean square error (MSE) or residual mean square, which is represented in option A as "the square root of MS_residual". This formula essentially captures the average distance that the observed values fall from the regression line.

Question 9

Which of the following is the standard error of estimate for a linear regression equation?​

Accepted Answer

The standard error of estimate for a linear regression equation is defined as the standard deviation of the residuals, which is estimated as the square root of SSE (sum of squared errors) divided by (n-2). Therefore, the correct formula for standard error of estimate is the square root of (SS_residual/(n-2)), which is option A.

Question 10

What is the standard error of estimate for a multiple regression equation with two predictor variables?​

Accepted Answer

The standard error of estimate for a multiple regression equation with two predictor variables is calculated using the formula: 
SE = sqrt(SSresidual/(n- k- 1))
where SSresidual is the sum of squared residuals, n is the sample size and k is the number of predictor variables. 
For a multiple regression equation with two predictor variables, k = 3 (two predictor variables and one intercept). Therefore, the correct formula for SE is:
SE = sqrt(SSresidual/(n-3))
This matches answer choice B.

Question 11

For a linear regression equation,the sum of the squared residuals can be computed directly by finding the difference between each Y and its predicted Y,then squaring the difference and adding the squared values.An alternative procedure is to calculate ____.​

Accepted Answer

The alternative procedure is to use the formula:
sum of squared residuals = (1 - r^2) * sum of squared Y deviations
where r is the correlation coefficient and sum of squared Y deviations is calculated as Σ(Y - Ŷ)^2. This formula is derived from the fact that r^2 represents the proportion of variability in Y that is explained by the regression equation, and (1 - r^2) represents the proportion of variability that is unexplained. The unexplained variability is captured by the sum of squared Y deviations. Therefore, multiplying (1 - r^2) by the sum of squared Y deviations gives the sum of squared residuals.

Question 12

An analysis of regression is used to test the significance of a linear regression equation based on a sample of n = 20 individuals.What are the df values for the F-ratio?​

Accepted Answer

The degrees of freedom for the F-ratio in a linear regression analysis are determined by the number of predictors and the sample size. For a simple linear regression (one predictor), the numerator df is 1 (for the predictor), and the denominator df is n - 2, where n is the sample size. With n = 20, the denominator df is 20 - 2 = 18.

Question 13

The multiple regression with two predictor variables is computed for a sample of n = 25 participants.What is the value for the degrees of freedom for the unpredicted portion of the Y-score variance,MS_residual?

Accepted Answer

The degrees of freedom for the unpredicted portion of the Y-score variance in multiple regression with two predictor variables is df = n - k - 1, where n is the sample size and k is the number of predictor variables (excluding the intercept term). In this case, k = 2, so df = 25 - 2 - 1 = 22. Therefore, the best choice is (C).

Question 14

A multiple regression equation is computed for a sample of n = 20 sets of X₁,X₂,and Y scores.If the significance of the regression equation is evaluated using an F-ratio,then the ratio would have degrees of freedom equal to ____.

Accepted Answer

The degrees of freedom for an F-ratio in multiple regression are determined by the number of predictors and the sample size. With 2 predictors (X1 and X2) and a sample size of n = 20, the degrees of freedom for the numerator (number of predictors) is 2, and for the denominator (n - number of predictors - 1), it is 20 - 2 - 1 = 17.

Question 15

The multiple regression with two predictor variables is computed for a sample of n = 25 participants.What is the value for the degrees of freedom for the predicted portion of the Y-score variance,MS_regression?

Accepted Answer

The degrees of freedom for the predicted portion of the Y-score variance in a multiple regression with two predictor variables is equal to the number of predictors, which is 2 in this case.

Question 16

A linear regression equation is calculated for a sample of n = 20 pairs of X and Y values.What would be the df value for the standard error of estimate?​

Accepted Answer

The degrees of freedom (df) for the standard error of estimate in a linear regression is calculated as n - 2, where n is the number of pairs. For n = 20, df = 20 - 2 = 18.

Question 17

A multiple regression equation with two predictor variables is calculated for a set of scores.If the constant values in the equation are b₁ = 2,b₂ = -3,and a = 7,then what Y value would be predicted for an individual with X₁ = 2 and X₂ = 4?

Accepted Answer

The predicted Y value is calculated as Y = a + b1*X1 + b2*X2 = 7 + 2*2 - 3*4 = 7 + 4 - 12 = -1.

Question 18

A linear regression equation is computed for a sample of n = 13 pairs of X and Y scores.For the analysis of regression testing the significance of the equation,what are the df values for the F-ratio?​

Accepted Answer

In a linear regression analysis, the degrees of freedom for the F-ratio are calculated as df1 = k - 1 and df2 = n - k, where k is the number of predictors (including the intercept) and n is the sample size. For a simple linear regression with one predictor and one intercept, k = 2. Therefore, df1 = 2 - 1 = 1, and df2 = 13 - 2 = 11, making the correct df values for the F-ratio 1, 11.

Question 19

Assuming that SS_Y is constant,which of the following correlations would have the largest SS_residual?

Accepted Answer

The residual sum of squares (SS_residual) is largest when the correlation (r) between the variables is weakest, as a weaker correlation indicates that the model explains less of the variance in the data, leaving more unexplained variance, which is captured by the residuals. Among the given options, r = -0.10 represents the weakest correlation.

Question 20

The Pearson correlation between X₁ and Y is r = 0.40 and SS_Y = 100.When a second variable,X₂,is added to the regression equation,we obtain R² = 0.25.How much additional variability is contributed by adding the second variable as a predictor compared to using X₁ alone?

Accepted Answer

The initial R² with X₁ alone is r² = 0.40² = 0.16. With X₂ added, R² = 0.25. The additional variability is 0.25 - 0.16 = 0.09. Since SS_Y = 100, the additional variability is 0.09 * 100 = 9 points.

Quiz 16: Introduction to Regression

A researcher who is evaluating the significance of a multiple regression equation with two predictor variables obtains an F-ratio with df = 2,24.How many individuals were needed to produce the sample data used to find the regression equation?​

For a linear regression calculated for a sample of n = 20 pairs of X and Y values,what is the value for degrees of freedom for the predicted portion of the Y-score variance,MSregression?​

A researcher who is evaluating the significance of a multiple regression equation with two predictor variables obtains an F-ratio with df = 2,18.How many individuals were needed to produce the sample data used to find the regression equation?​

A multiple regression equation with two predictor variables produces R2 = 0.10.What portion of the variability for the Y scores is predicted by the equation?​

The Pearson correlation between X1 and Y is r = 0.50 and the correlation between X2 and Y is r = 0.20.How much variance for the Y scores is predicted by using both X1 and X2 as predictor variables in a multiple regression equation?​

For a linear regression calculated for a sample of n = 20 pairs of X and Y values,what is the value for degrees of freedom for the unpredicted portion of the Y-score variance,MSresidual?​

The distance between the Y value in the data and the Y value predicted from the regression equation is known as the residual.What is the value for the sum of the squared residuals?​

For either linear regression or multiple regression,the standard error of estimate can be computed as ____.​

Which of the following is the standard error of estimate for a linear regression equation?​

What is the standard error of estimate for a multiple regression equation with two predictor variables?​

For a linear regression equation,the sum of the squared residuals can be computed directly by finding the difference between each Y and its predicted Y,then squaring the difference and adding the squared values.An alternative procedure is to calculate ____.​

An analysis of regression is used to test the significance of a linear regression equation based on a sample of n = 20 individuals.What are the df values for the F-ratio?​

The multiple regression with two predictor variables is computed for a sample of n = 25 participants.What is the value for the degrees of freedom for the unpredicted portion of the Y-score variance,MSresidual?​

A multiple regression equation is computed for a sample of n = 20 sets of X1,X2,and Y scores.If the significance of the regression equation is evaluated using an F-ratio,then the ratio would have degrees of freedom equal to ____.​

The multiple regression with two predictor variables is computed for a sample of n = 25 participants.What is the value for the degrees of freedom for the predicted portion of the Y-score variance,MSregression?​

A linear regression equation is calculated for a sample of n = 20 pairs of X and Y values.What would be the df value for the standard error of estimate?​

A multiple regression equation with two predictor variables is calculated for a set of scores.If the constant values in the equation are b1 = 2,b2 = -3,and a = 7,then what Y value would be predicted for an individual with X1 = 2 and X2 = 4?​

A linear regression equation is computed for a sample of n = 13 pairs of X and Y scores.For the analysis of regression testing the significance of the equation,what are the df values for the F-ratio?​

​Assuming that SSY is constant,which of the following correlations would have the largest SSresidual?

The Pearson correlation between X1 and Y is r = 0.40 and SSY = 100.When a second variable,X2,is added to the regression equation,we obtain R2 = 0.25.How much additional variability is contributed by adding the second variable as a predictor compared to using X1 alone?​

A researcher who is evaluating the significance of a multiple regression equation with two predictor variables obtains an F-ratio with df = 2,24.How many individuals were needed to produce the sample data used to find the regression equation?

For a linear regression calculated for a sample of n = 20 pairs of X and Y values,what is the value for degrees of freedom for the predicted portion of the Y-score variance,MS_regression?

A researcher who is evaluating the significance of a multiple regression equation with two predictor variables obtains an F-ratio with df = 2,18.How many individuals were needed to produce the sample data used to find the regression equation?

A multiple regression equation with two predictor variables produces R² = 0.10.What portion of the variability for the Y scores is predicted by the equation?

The Pearson correlation between X₁ and Y is r = 0.50 and the correlation between X₂ and Y is r = 0.20.How much variance for the Y scores is predicted by using both X₁ and X₂ as predictor variables in a multiple regression equation?

For a linear regression calculated for a sample of n = 20 pairs of X and Y values,what is the value for degrees of freedom for the unpredicted portion of the Y-score variance,MS_residual?

The distance between the Y value in the data and the Y value predicted from the regression equation is known as the residual.What is the value for the sum of the squared residuals?

For either linear regression or multiple regression,the standard error of estimate can be computed as ____.

Which of the following is the standard error of estimate for a linear regression equation?

What is the standard error of estimate for a multiple regression equation with two predictor variables?

For a linear regression equation,the sum of the squared residuals can be computed directly by finding the difference between each Y and its predicted Y,then squaring the difference and adding the squared values.An alternative procedure is to calculate ____.

An analysis of regression is used to test the significance of a linear regression equation based on a sample of n = 20 individuals.What are the df values for the F-ratio?