A line that is used to represent the linear trend in the data is called the line of best fit. When finding the error of the line of best fit, which statement holds true?
A. If the error is less than one, the line of best fit is the linear equation of the data.
B. The smaller the error, the better the fit.
C. If the error is greater than one, there is no correlation to the line of best fit.
D. The larger the error, the better the fit.
Answer (1)
The problem explores the relationship between the error of a line of best fit and the quality of the fit.
The error represents the difference between actual data points and values predicted by the line.
A smaller error indicates a better fit, meaning the line is a more accurate representation of the data.
A larger error indicates a poorer fit.
Therefore, the correct statement is: The smaller the error, the better the fit. $\boxed{b}
Explanation
Understanding the Problem The problem describes the concept of a line of best fit and asks about the relationship between the error of the line of best fit and how well it represents the data. We need to determine which of the given statements accurately describes this relationship.
Analyzing the Statements Let's analyze each statement:
a. If the error is less than one, the line of best fit is the linear equation of the data. This is generally false. A small error doesn't guarantee the line perfectly represents the data, it just means the data points are relatively close to the line.
b. The smaller the error, the better the fit. This is generally true. Smaller errors indicate the line is closer to the data points, meaning it's a better representation of the data.
c. If the error is greater than one, there is no correlation to the line of best fit. This is generally false. An error greater than one doesn't necessarily mean there's no correlation, just that the fit isn't perfect. There can still be a linear trend, even if the error is relatively large.
d. The larger the error, the better the fit. This is false. Larger errors indicate a poor fit, as the data points are further away from the line.
Selecting the Correct Statement Based on the analysis, the statement that holds true is:
b. The smaller the error, the better the fit.
Final Answer Therefore, the correct answer is b.
Examples
Imagine you're trying to predict a student's test score based on the number of hours they study. The line of best fit helps you make this prediction. The error represents how far off your prediction is from the student's actual score. If the error is small, your predictions are more accurate, and the line of best fit is a good representation of the relationship between study hours and test scores. This concept is used in various fields like finance to predict stock prices or in science to analyze experimental data.
