The Covariance and Variance

Covariance (cov) and variance (var) are statistical measures used to understand the relationship between two random variables. Covariance measures how two variables vary together, while variance measures the dispersion of a single variable.

Covariance (cov(X, Y))

Covariance measures the extent to which two variables vary together. It indicates the direction of the relationship between the variables (positive or negative) and the strength of the relationship.

In this question, cov(X, Y) is given as 10.

Variance (var(X) and var(Y))

Variance measures the dispersion or spread of a single variable. It quantifies how much the values of a variable deviate from the mean.

In this question, var(X) is given as 6.25 and var(Y) is given as 31.36.

Calculating the Correlation Coefficient (ρ)

The correlation coefficient (ρ) is a standardized measure of the linear relationship between two variables. It is calculated as the ratio of the covariance to the square root of the product of the variances.

ρ = cov(X, Y) / √(var(X) * var(Y))

Substituting the given values, we have:

ρ = 10 / √(6.25 * 31.36)

ρ = 10 / √195.76

ρ = 10 / 13.986

ρ ≈ 0.715

Interpreting the Correlation Coefficient

The correlation coefficient ranges from -1 to 1. A value of 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship.

In this question, the correlation coefficient (ρ) is approximately 0.715. Since it is positive and not close to 1 or -1, it indicates a moderately strong positive linear relationship between X and Y.

Interpreting the Correct Answer

The correct answer is option A: 5/7.

The correlation coefficient (ρ) is approximately 0.715, which can be expressed as 5/7. Therefore, option A is the correct answer.

Note: The correlation coefficient measures the strength and direction of the linear relationship between two variables. It does not provide information about the actual values or magnitudes of the variables.