Final answer:
The least-squares regression line minimizes the sum of squared residuals, is the line that best fits the data, and can be used for predictions within the data set, but it does not always pass through the origin.
Step-by-step explanation:
The question you've asked relates to the characteristics of the least-squares regression line in statistics. Here are the truths about the statements you've provided:
- True: The least-squares regression line minimizes the sum of the squared residuals. This means it makes the overall distance of the data points from the line as small as possible, in terms of their vertical deviations, which are squared to ensure that positive and negative deviations don't cancel each other out.
- False: The least-squares regression line does not always pass through the origin (0,0). It only passes through the origin if the relationship between the variables dictates that when x is 0, y is also 0, and the nature of the data supports this.
- True: The least-squares regression line is the line that best fits the data points on a scatter plot. It is the line of best fit found by the least-squares method.
- True: The least-squares regression line can be used to predict the value of the dependent variable based on the independent variable within the range of the original data set.
When constructing a regression line, we use a formula of the type ý = a + bx where ý is the predicted value of the dependent variable (y) for a given x, a is the y-intercept, and b is the slope. The formula itself is derived by using methods such as calculus to minimize the sum of squared errors.