The coefficient of determination measures:
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A. B. C. D.D
The coefficient of determination, also known as the R-square of the regression, measures the amount of variance of the dependent variable explained by the independent variable.
The coefficient of determination, also known as R-squared (R²), is a statistical measure used in regression analysis to assess the goodness of fit of a regression model. It represents the proportion of the variance in the dependent variable that is predictable or explained by the independent variable(s) in the model.
Option D, "The amount of variance of the dependent variable explained by the independent variable," correctly describes the coefficient of determination. Let's break down this explanation further:
Variance is a statistical measure that quantifies the dispersion or spread of a set of data points. In the context of regression analysis, variance refers to the variability or fluctuations in the values of the dependent variable.
When we perform a regression analysis, we are trying to create a mathematical relationship or model between the dependent variable and one or more independent variables. The dependent variable is the variable we are trying to predict or explain, while the independent variable(s) are the factors or variables we believe have an impact on the dependent variable.
The coefficient of determination tells us the proportion of the total variance in the dependent variable that can be attributed to the independent variable(s) in the regression model. In other words, it quantifies how much of the variation in the dependent variable is accounted for or "explained" by the independent variable(s) in the model.
The coefficient of determination, denoted by R², takes values between 0 and 1. A value of 0 indicates that none of the variability in the dependent variable is explained by the independent variable(s), while a value of 1 indicates that all of the variability in the dependent variable is explained by the independent variable(s). Therefore, the closer the R² value is to 1, the better the regression model fits the data and the more effectively the independent variable(s) explain the dependent variable.
Option A, "The percentage change in the dependent variable caused by a 1% change in the independent variable," refers to the concept of elasticity, not the coefficient of determination. Elasticity measures the responsiveness or sensitivity of the dependent variable to changes in the independent variable(s), usually expressed in percentage terms.
Option B, "The degree of linear association between the dependent and the independent variables," refers to the correlation coefficient, not the coefficient of determination. The correlation coefficient measures the strength and direction of the linear relationship between two variables but does not provide information about the amount of variance explained.
Option C, "The slope of the regression line," refers to the coefficient(s) of the independent variable(s) in the regression equation, not the coefficient of determination. The slope represents the change in the dependent variable associated with a unit change in the independent variable, while the coefficient of determination provides an overall measure of the explanatory power of the independent variable(s).
In summary, the coefficient of determination (R²) measures the amount of variance in the dependent variable that can be explained by the independent variable(s) in a regression model. It is a valuable tool for assessing the goodness of fit of the model and evaluating the strength of the relationship between the variables.