Question

True/false: If the correlation is \(.8,\) then \(40 \%\) of the variance is explained.

Short answer

False, because 64% of the variance is explained.

Step-by-step solution

Understanding Correlation

First, let's understand the term 'correlation'. Correlation, denoted as \( r \), measures the strength and direction of a linear relationship between two variables. Here, it is given as \( r = 0.8 \).

Calculate Variance Explained

To find the variance explained by the correlation, we use the square of the correlation coefficient \( r \). The portion of variance explained is \( r^2 \).

Square the Correlation Coefficient

Calculate \( r^2 \) using \( r = 0.8 \): \[ r^2 = 0.8^2 = 0.64 \]

Compare the Calculated Variance to Given Information

The result from Step 3 shows that \( 64\% \) of the variance is explained because \( r^2 = 0.64 \). The problem statement claims that \( 40\% \) of the variance is explained. We need to check if these two numbers match.

Conclusion

Since \( 64\% \) (from the calculation) does not equal \( 40\%\) (as claimed in the statement), the statement is false.

Key concepts

Correlation Coefficient

The correlation coefficient is a key concept in statistics that indicates the strength and direction of a linear relationship between two variables. This value, denoted by the symbol \( r \), ranges from \(-1\) to \(1\). A correlation coefficient of \(1\) signifies a perfect positive linear relationship, while \(-1\) denotes a perfect negative linear relationship. If the correlation coefficient is \(0\), it indicates no linear relationship between the variables.

When analyzing the correlation coefficient of \(0.8\) as in our exercise, it suggests a strong positive linear relationship between the variables involved. This means as one variable increases, the other variable tends to increase as well. However, while the correlation coefficient tells us about the relationship's strength and direction, it doesn’t tell us anything about causation. For example, factors outside the linear relationship might also influence the variables.

Understanding and interpreting the correlation coefficient correctly is crucial as it aids in predicting one variable using another. This prediction accuracy is explored further through the concept of variance explained.

Variance Explained

Variance explained is a statistical measure that illustrates how much of the variation in one variable can be predicted from another variable. By squaring the correlation coefficient \( r \), we find the proportion of variance explained, known as \( r^2 \). In the context of the exercise, if \( r = 0.8 \), then \( r^2 = 0.8^2 = 0.64 \) or \( 64\% \).

This means that \( 64\% \) of the variability in one variable is accounted for by its linear relationship with the other variable. The higher this percentage, the stronger the explanatory power of the correlation. It's important to remember that even if a significant portion of variance is explained, not all of it is. This leaves room for other factors or variables, not captured by \( r^2 \), that might affect the variable in question.

The mistake in the exercise was assuming \( 40\% \) of the variance was explained, which is incorrect as proven by the calculation. Recognizing this discrepancy emphasizes the importance of correctly computing and interpreting \( r^2 \).

Linear Relationship

A linear relationship is characterized by a straight-line connection between two variables on a graph. The strength of this linear relationship can be understood through the correlation coefficient \( r \). In the exercise, the given correlation coefficient of \( 0.8 \) implies a strong, positive linear relationship, indicating that as one variable increases, the other does too, at a predictable rate.

Linear relationships are particularly useful because they allow us to create models that can predict or infer values. This predictability stems from the assumption that any change in one variable is systematically mirrored by the other. However, it’s key to remember that correlation and linearity do not imply causation. External factors could be at play even if the correlation appears robust.

In analyzing data, it's crucial to visually assess scatter plots to confirm the linear nature of relationships, complemented by statistical indicators like the correlation coefficient. This holistic approach ensures a comprehensive understanding of the relationship dynamics between variables.