Final answer:
The least squares method is used in regression analysis to find the equation of the line of best fit by minimizing the sum of squared errors. The coefficient of determination (r-squared) generated from this method indicates the fit of the regression line to the data. The slope 'b' of the least squares line suggests the change in the dependent variable per unit increase in the independent variable.
Step-by-step explanation:
Understanding the Least Squares Method
The least squares method is a statistical technique used in regression analysis to determine the line of best fit for a set of data. The goal is to minimize the sum of the squares of the residuals—the differences between observed and predicted values. To find the estimated regression equation, one calculates the line ý = a + bx, where 'a' is the y-intercept and 'b' is the slope.
When calculating this line, the sum of squared errors (SSE) is minimized, ensuring that any other line would have a higher SSE than the chosen one. The coefficient of determination, symbolized as 'r-squared', utilizes the least squares method by measuring how well the regression line approximates the real data points. An 'r-squared' value closer to 1 indicates a line that fits the data well.
To find the estimated values, such as life expectancy or CPI for a specific year, one would plug the independent variable into the regression equation. Regarding the interpretation of the slope 'b', it represents the estimated change in the dependent variable for each unit increase in the independent variable. It's important to note that the least squares line is most accurate within the range of the data provided and should be used cautiously for extrapolation beyond that range.