Final answer:
Regression lines model the relationship between an independent and a dependent variable, can be used for predictions, do not always pass through the origin, and can be linear or nonlinear.
Step-by-step explanation:
Understanding Regression Lines
Regression lines are powerful tools in statistics used to model the relationship between two variables, known as the independent and dependent variables. Statement 1 is true: They are indeed used to model relationships, usually between an independent variable (like time, in years) and a dependent variable (such as the number of flu cases).
Statement 2 is also true: Regression lines can be utilized to make predictions within a data set. For instance, using the least-squares regression line, which minimizes the sum of squared errors (SSE), one can predict unknown values of a dependent variable for given values of the independent variable.
Statement 3 is false: Regression lines do not always pass through the origin unless the relationship dictates that the y-intercept is zero (when x=0, y should also be 0).
Statement 4 is true: There can be linear or nonlinear regression lines. Linear regression involves a straight-line relationship, typically expressed as y = a + bx, whereas nonlinear regression might involve curvilinear or more complex relationships between variables.
The line of best fit is determined through methods such as the least-squares and is used for making predictions, though caution should be exercised not to predict beyond the scope of the data set.