Least-squares regression line

Definition

A least-squares regression line is a straight line that best fits the data points on a scatter plot by minimizing the sum of the squares of the vertical distances (residuals) between observed values and the line.

5 Must Know Facts For Your Next Test

1. The equation of the least-squares regression line is given by $\hat{y} = b_0 + b_1x$, where $b_0$ is the y-intercept and $b_1$ is the slope.
2. The slope $b_1$ represents the average change in the dependent variable for each one-unit change in the independent variable.
3. The y-intercept $b_0$ represents the expected value of the dependent variable when the independent variable equals zero.
4. The correlation coefficient, denoted as $r$, helps to determine how well data points fit a linear relationship; it ranges from -1 to 1.
5. Outliers can significantly affect the position and slope of a least-squares regression line.

Review Questions

• What does the slope ($b_1$) represent in a least-squares regression line?
• How is it determined whether a least-squares regression line fits the data well?
• Why are residuals important in calculating a least-squares regression line?

Related terms

Correlation Coefficient: A statistical measure denoted as $r$ that describes the strength and direction of a linear relationship between two variables.

Residual: The difference between an observed value and its corresponding predicted value on a regression line.

Scatter Plot: A graph used to visually display and assess relationships between two quantitative variables, with one variable plotted along each axis.