For a linear regression with one explanatory variable, which statement is true?

The statement that R² is the ratio of the regression sum of squares to the total sum of squares is correct. In a linear regression context, R², or the coefficient of determination, quantifies how much of the variability in the dependent variable can be explained by the independent variable. It is calculated as the proportion of variance explained by the regression model relative to the total variance in the data.

Mathematically, R² is defined as:

[ R^2 = \frac{\text{Regression Sum of Squares}}{\text{Total Sum of Squares}} = 1 - \frac{\text{Residual Sum of Squares}}{\text{Total Sum of Squares}} ]

This formulation captures the relationship between the explained variation and the total variation in the dataset. A higher R² value indicates a better fit of the model to the data, meaning more variance is explained by the model.

The other options do not accurately represent key concepts involved in regression analysis. For instance, while R² is related to the correlation coefficient, it is distinct from it; they reflect different things even though they are related in simple linear regression. The statement about the standard error refers to variability across all predicted values or models rather than for just a single observation, highlighting an