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.

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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.

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.

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.

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.

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.

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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.