Least Squares Regression Equation Calculation by Hand

Calculate the least squares regression equation $\hat{y} = \beta_0 + \beta_1 x$ for the given small data set by hand. Find the estimates for the intercept $\beta_0$ and the slope $\beta_1$.

When calculating the least squares regression equation by hand for a small data set, it's important to grasp the fundamental aspects of linear regression. Linear regression is a powerful statistical method used to model and analyze the relationships between variables. In this particular case, we focus on finding the best-fitting line through the points in the data set, which is achieved by minimizing the squared differences (or errors) between the observed values and the values predicted by our line. This approach is known as the method of least squares, which forms the foundation for much of regression analysis.

The key idea in least squares regression is to establish a linear relationship between the independent variable (often denoted as x) and the dependent variable (denoted as y). The equation takes the form $\hat{y} = \beta_0 + \beta_1 x$, where $\beta_0$ represents the y-intercept and $\beta_1$ represents the slope of the line. The slope, $\beta_1$, provides insights into the nature of the relationship between x and y; specifically, it indicates the change in the dependent variable for a one-unit change in the independent variable. Calculating these parameters involves using formulas derived from the principles of minimizing the sum of the squared residuals.

Understanding this concept allows students to appreciate the simplicity and elegance of linear regression in exploring relationships within data sets, especially before delving into software-driven analysis. This exercise solidifies the conceptual groundwork necessary for more complex applications in statistics, such as multiple regression, and enables students to critically engage with statistical outputs and reports.