Question 1

The statistical significance of the linear relationship between X and Y may be tested by examining the hypotheses: H₀: β₁ ≠0; H₁: β₁=0.

Accepted Answer

The correct null hypothesis for testing the statistical significance of the linear relationship between X and Y (where β1 represents the slope coefficient in a simple linear regression) is H0: β1 = 0, not H0: β1 ≠ 0. The alternative hypothesis is typically H1: β1 ≠ 0, indicating there is a significant linear relationship.

Question 2

If a variable explains a significant proportion of the residual variation, it should be considered for inclusion in the regression equation.

Accepted Answer

If a variable explains a significant proportion of the residual variation, it means that it has a strong relationship with the dependent variable and is contributing to the predictive power of the regression model. Therefore, it should be considered for inclusion in the equation.

Question 3

The standard error of estimate, SEE, is the standard deviation of the actual Y values from the predicted Ŷ values.

Accepted Answer

The statement correctly defines the standard error of estimate (SEE). The SEE measures the spread of the actual Y values around the regression line and indicates how well the regression equation fits the data. It is calculated as the square root of the residual sum of squares (RSS) divided by the degrees of freedom (df), where RSS is the sum of squared residuals between Y and Ŷ and df is the number of observations minus the number of predictors minus 1.

Question 4

The formula for the coefficient of determination is r² = SS_reg/ SS_y.

Accepted Answer

The answer of The formula for the coefficient of determination...

Question 5

R²cannot decrease as more independent variables are added to the regression equation.

Accepted Answer

The sum of squares of the regression (SSR) always increases or remains the same as more independent variables are added. Therefore, the residual sum of squares (RSS) will decrease or remain the same, resulting in the R-squared value (S1S1P0) not decreasing as more independent variables are added.

Question 6

A residual is the difference between the observed value of Y_i and the value predicted by the regression equation, Ŷ_i.

Accepted Answer

This is the definition of a residual in a regression analysis.

Question 7

If an examination of the residuals indicates that the assumptions underlying linear regression are not met, the researcher can transform the variables in an attempt to satisfy the assumptions.

Accepted Answer

Yes, this is true. Transforming variables (such as taking logarithms or square roots) can often improve the normality, linearity, and equal variance assumptions needed for linear regression. However, the researcher should also consider alternative modeling approaches if transformations do not adequately address the issues with the residuals.

Question 8

The coefficient of multiple determination is adjusted for the number of dependent variables and the sample size to account for diminishing returns.

Accepted Answer

The coefficient of multiple determination is adjusted for the number of independent variables and the sample size to account for the diminishing returns of adding more variables, not dependent variables.

Question 9

Standardized variables have a mean of 1 and a variance of zero.

Accepted Answer

Standardized variables have a mean of 0 and a variance of 1. Zero variance means that all observations are the same and this is not a useful variable.

Question 10

The vertical distance from a point to the regression line is the squared error, e²^.

Accepted Answer

The vertical distance from a point to the regression line is the residual, not the squared error. The squared error is the sum of the squared residuals.

Question 11

The hypotheses for the test for significance of the coefficient of determination are: H₀: R²_pop= 0 ; H₁: R²_pop> 0.

Accepted Answer

The hypotheses for the test for significance of the coefficient of determination are typically in the form of H₀: R²_pop= 0 (null hypothesis) and H₁: R²_pop> 0 (alternative hypothesis). This means that the answer is A.

Question 12

The term beta coefficient or beta weight is used to denote the standardized regression coefficient.

Accepted Answer

The beta coefficient is a standardized regression coefficient that reflects the change in the dependent variable for each one-unit change in the independent variable. It is also referred to as a beta weight.

Question 13

When fitting a straight line to a scattergram, the best-fitting line is called the regression line.

Accepted Answer

The statement is true. The regression line is the line that minimizes the distance between the points in the scattergram and the line.

Question 14

In multiple regression, if the overall null hypothesis is rejected, we know which specific coefficients (β_is)are nonzero.

Accepted Answer

Rejecting the overall null hypothesis in multiple regression only tells us that at least one of the predictors is significantly related to the outcome variable. It does not specify which predictors are significant.

Question 15

The multiple correlation coefficient, R, can also be viewed as the simple correlation coefficient, r, between Y and Ŷ.

Accepted Answer

This is true, as the multiple correlation coefficient measures the relationship between the independent variables and the dependent variable, while the simple correlation coefficient measures the relationship between two individual variables. However, in the case of multiple regression, the multiple correlation coefficient can also be represented as the simple correlation coefficient between the dependent variable and the predicted values (Ŷ) based on the independent variables.

Question 16

Stepwise procedures result in regression equations that are optimal, in the sense of producing the largest R², for a given number of predictors.

Accepted Answer

Stepwise procedures do not guarantee an optimal regression equation in terms of R-squared for a given number of predictors; they are heuristic methods that may overlook the best model due to their sequential nature of adding or removing variables based on specific criteria at each step.

Question 17

The general form of the multiple regression model is: Y= β₀+ β₁ X₁+β₂ X₂+β₃X₃+ ....+ β_kX_{k +}e

Accepted Answer

This is the general form of the multiple regression model, where Y is the dependent variable, B0 is the intercept, B1 to Bk are the coefficients of the independent variables, and e is the error term.

Question 18

The purpose of stepwise regression is to select, from a large number of predictor variables, a small subset of variables that account for most of the variation in the dependent or criterion variable.

Accepted Answer

This is the main purpose of stepwise regression.

Question 19

The standard error of estimate, SEE, may be interpreted as a kind of average residual or average error in predicting Y from the regression equation.

Accepted Answer

This is a correct interpretation of the standard error of estimate.

Question 20

When there are a large number of independent variables and the researcher suspects that not all of them are significant, stepwise regression should be used.

Accepted Answer

Stepwise regression is a useful tool when dealing with a large number of independent variables, as it automatically selects the most significant variables to include in the final model. This can save time and effort in the data analysis process.

Quiz 17: Correlation and Regression

The statistical significance of the linear relationship between X and Y may be tested by examining the hypotheses: H0: β1 ≠ 0; H1: β1 = 0.