Final answer:
Residuals in multiple regression analysis are used to measure the difference between the actual and predicted values of the dependent variable, assess the regression model's fit, and identify outliers and potential improvements in the model.
Step-by-step explanation:
In multiple regression analysis, residuals (Y - Y') are used to measure the difference between the actual values of the dependent variable and the values predicted by the regression model. This difference is also known as the error of the prediction. By examining residuals, one can assess how well the regression line fits the data. A regression line that fits the data well will have smaller residuals, indicating that the predicted values are close to the actual values. Conversely, large residuals suggest that the model may not be capturing some of the variability in the data, possibly due to a non-linear relationship or outliers. Residual analysis is essential for checking the assumptions of linear regression, identifying outliers, and assessing the potential for improving the model.
The sum of squared errors (SSE) is minimized to find the least-squares regression line, which is considered the best fit for a given set of data. This process ensures that the overall distance from the data points to the regression line is as small as possible. It's crucial to note that regression lines should only be used to make predictions within the range of the given data set and not beyond it.
Key aspects of regression analysis include the slope and y-intercept of the regression line, which provide valuable information about the relationship between the independent and dependent variables. Additionally, the standard deviation of residuals can be used to estimate the population standard deviation of the dependent variable, which helps in evaluating the model's precision.