The equation ŷ = -7.61 + 0.19x is called the least-squares regression line because it:
1. **Minimizes the sum of the squared differences:** The least-squares regression line is obtained by finding the line that minimizes the sum of the squared vertical distances between each data point and the line itself. This minimization process is what makes it the "least-squares" regression line.
2. **Best fits the data points:** By minimizing the sum of the squared errors, the least-squares regression line provides the best-fitting line through a set of data points. It represents the line that best represents the relationship between the independent variable (x) and the dependent variable (ŷ) based on the given data.
3. **Predicts the dependent variable:** Once the least-squares regression line is determined, it can be used to predict the value of the dependent variable (ŷ) for a given value of the independent variable (x). This predictive capability makes it a valuable tool in statistical analysis and forecasting.
4. **Provides a measure of goodness-of-fit:** The equation of the least-squares regression line includes the intercept (-7.61) and the slope (0.19), which together describe the relationship between the variables. The accuracy of the fit can be assessed by examining the residuals (the differences between the actual data points and the predicted values on the line).