Question

True or False The correlation coefficient is a measure of the strength of a linear relation between two variables and must lie between -1 and 1 , inclusive.

Short answer

True

Step-by-step solution

Understanding the Correlation Coefficient

The correlation coefficient, often denoted as \(\rho\) or \(r\), is a statistical measure that calculates the strength and direction of a linear relationship between two variables.

Range of the Correlation Coefficient

The value of the correlation coefficient must lie between \(-1\) and \(1\), inclusive. This means that the correlation coefficient \(r\) cannot be less than \(-1\) or greater than \(1\).

Interpreting the Range

A correlation coefficient of \(+1\) indicates a perfect positive linear relationship, \(-1\) indicates a perfect negative linear relationship, and \(0\) indicates no linear relationship.

Conclusion

Given the definitions and properties of the correlation coefficient, it is correct to say that it must lie between \(-1\) and \(1\) and measures the strength of a linear relation between two variables. Therefore, the statement is true.

Key concepts

linear relationship

A linear relationship describes how two variables change together in a consistent way. Imagine you're tracking your study hours and exam scores. If increasing your study hours always boosts your exam score by a fixed amount, you have a linear relationship.

This relationship can be visualized in a graph as a straight line. In simpler terms, if one variable goes up, the other either goes up (positive direction) or down (negative direction) consistently.

For instance, in our study hours example, if every extra hour studied results in 10 more points on your exam, that's a linear relationship. We use the correlation coefficient to measure the strength and direction of this linear relationship.

statistical measure

The correlation coefficient, often represented as \(r\) or \(\rho\), is a statistical measure. It helps us understand how closely two variables move concerning each other. When you hear 'statistical measure,' think about how we can use numbers to summarize and give meaning to data.

The correlation coefficient specifically focuses on linear relationships. Statisticians love it because it gives them a single number that summarizes the strength and direction of a relationship.

An \(r\) value close to +1 or -1 means the variables have a strong linear relationship. If \(r\) is close to 0, the relationship is weak or non-linear. This way, we can quickly judge how much one variable changes when the other one does.

range of correlation coefficient

The range of the correlation coefficient is from -1 to +1, inclusive. These values are boundaries showing the extent of a linear relationship. Here's what the values mean:

• \(+1\): Indicates a perfect positive linear relationship. When one variable increases, the other increases by a consistent amount.
• \(-1\): Indicates a perfect negative linear relationship. When one variable increases, the other decreases by a consistent amount.
• \(0\): Indicates no linear relationship. The variables don't affect each other linearly.

Values closer to -1 or +1 mean stronger relationships, while values near 0 suggest weaker relationships. This range is crucial for interpreting the strength and the direction of the relationship between two variables.