What do the y-coordinates on the least-squares regression line represent?

Verified step by step guidance

1. Understand that the least-squares regression line is a line that best fits the data points in a scatterplot by minimizing the sum of the squared vertical distances between the observed points and the line.
2. Recall that the equation of the least-squares regression line is typically written as \(y = \hat{y} = b_0 + b_1 x\), where \(b_0\) is the y-intercept and \(b_1\) is the slope.
3. Recognize that for any given value of the explanatory variable \(x\), the corresponding y-coordinate on the regression line, denoted \(\hat{y}\), is the predicted or estimated value of the response variable \(y\).
4. Interpret the y-coordinate \(\hat{y}\) as the expected value of \(y\) given \(x\), meaning it represents the average or typical value of \(y\) when \(x\) is fixed at a certain value.
5. Therefore, the y-coordinates on the least-squares regression line represent the predicted values of the response variable based on the linear relationship modeled by the regression.

Key Concepts

Least-Squares Regression Line

The least-squares regression line is a straight line that best fits a set of data points by minimizing the sum of the squared vertical distances (residuals) between observed values and the line. It models the relationship between an independent variable (x) and a dependent variable (y).

Predicted Values

The y-coordinates on the regression line represent predicted values of the dependent variable for given values of the independent variable. These predictions estimate the average expected outcome based on the linear relationship captured by the regression line.

Interpretation of Coordinates in Regression

In regression analysis, each point on the line corresponds to a specific x-value and its predicted y-value, showing the expected response. These y-coordinates help understand how changes in the independent variable influence the dependent variable on average.