Final answer:
The linear function Y = a + bX is used in linear regression to model the relationship between an independent variable X and a dependent variable Y, where 'a' is the y-intercept and 'b' is the slope. This model is useful for making predictions within the range of the data sample.
Step-by-step explanation:
The equation Y = a + bX represents a linear function used in linear regression to estimate the relationship between an independent variable X and a dependent variable Y. The parameter a in the equation represents the y-intercept, which is the value of Y when X is zero.
The parameter b reflects the slope of the line, indicating the rate at which Y changes when X changes. If there is a significant linear relationship, it means that we can use the regression line to model the relationship in the population. For instance, if X represents hours studied and Y represents test scores, a positive slope (b) suggests that more hours studied are associated with higher test scores.
Linear regression is commonly used to draw a regression line, or line of best fit, based on the least-squares method, which minimizes the sum of squared errors (SSE) between the actual Y values and the predicted Y values on the line. This regression line can be used to make predictions about Y based on given values of X within the range of the data used to create the model, but it is not appropriate for extrapolating predictions outside that range.