The _____ _____ _____, R^2, quantifies the proportion of total variation in the response variable explained by the least-squares regression line.

Verified step by step guidance

1. Identify the term described: The problem refers to a statistic that measures how well the regression line explains the variation in the response variable.
2. Recall that in regression analysis, the coefficient of determination, denoted as \(R^2\), quantifies the proportion of the total variation in the response variable that is explained by the regression model.
3. Understand that \(R^2\) is calculated as the ratio of the explained variation to the total variation, which can be expressed as \(R^2 = \frac{SS_{regression}}{SS_{total}}\), where \(SS_{regression}\) is the sum of squares due to regression and \(SS_{total}\) is the total sum of squares.
4. Recognize that the missing phrase is 'coefficient of determination', which is the formal name for \(R^2\).
5. Therefore, the complete phrase is: 'The coefficient of determination, \(R^2\), quantifies the proportion of total variation in the response variable explained by the least-squares regression line.'

Key Concepts

Coefficient of Determination (R²)

R² measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s) using the regression model. It ranges from 0 to 1, where higher values indicate a better fit of the model to the data.

Least-Squares Regression Line

This is the line that minimizes the sum of the squared differences between observed values and predicted values. It provides the best linear fit to the data, allowing prediction of the response variable based on the explanatory variable.

Total Variation in the Response Variable

Total variation refers to the overall spread or variability of the observed data points around their mean. Understanding this helps in assessing how much of this variability is explained by the regression model versus unexplained error.