3. What does the sample correlation coefficient r measure? Which value indicates a stronger correlation: r =0.918 or r =- 0.932? Explain your reasoning.

Verified step by step guidance

1. The sample correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where values close to -1 or 1 indicate a strong linear relationship, and values near 0 indicate a weak or no linear relationship.
2. A positive value of r (e.g., r = 0.918) indicates a positive linear relationship, meaning as one variable increases, the other tends to increase as well. A negative value of r (e.g., r = -0.932) indicates a negative linear relationship, meaning as one variable increases, the other tends to decrease.
3. To determine which value indicates a stronger correlation, focus on the magnitude of r (the absolute value), not the sign. The closer the absolute value of r is to 1, the stronger the linear relationship.
4. Compare the absolute values of r = 0.918 and r = -0.932. The absolute value of r = 0.918 is |0.918| = 0.918, and the absolute value of r = -0.932 is |-0.932| = 0.932.
5. Since 0.932 is greater than 0.918, r = -0.932 indicates a stronger correlation than r = 0.918, even though it is negative. The sign only tells the direction of the relationship, not its strength.

Key Concepts

Sample Correlation Coefficient (r)

The sample correlation coefficient, denoted as r, quantifies the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where values close to 1 indicate a strong positive correlation, values close to -1 indicate a strong negative correlation, and values around 0 suggest no correlation. Understanding r is crucial for interpreting how changes in one variable may relate to changes in another.

Strength of Correlation

The strength of correlation refers to how closely the data points cluster around a line of best fit. A higher absolute value of r signifies a stronger correlation, regardless of the sign. For instance, r = 0.918 and r = -0.932 both indicate strong correlations, but the latter is stronger due to its absolute value being closer to -1, demonstrating a more pronounced relationship.

Positive vs. Negative Correlation

Correlation can be positive or negative, indicating the direction of the relationship between variables. A positive correlation (r > 0) means that as one variable increases, the other also tends to increase, while a negative correlation (r < 0) indicates that as one variable increases, the other tends to decrease. Understanding this distinction is essential for interpreting the implications of the correlation coefficient in real-world contexts.