Explanation

The correlation coefficient is calculated using the formula:

$\rho_{X,Y} = \frac{\text{Cov}(X,Y)}{\sigma_X \cdot \sigma_Y}$

From the covariance matrix:

• Variance of X = 650, so standard deviation σ_X = √650 = 25.50
• Variance of Y = 450, so standard deviation σ_Y = √450 = 21.21
• Covariance between X and Y = 120

Now calculate: $\rho_{X,Y} = \frac{120}{25.50 \times 21.21} = \frac{120}{540.855} = 0.2218 \approx 0.22$

Therefore, the correlation coefficient is approximately 0.22.

Key points:

1. The diagonal elements of a covariance matrix represent variances
2. Off-diagonal elements represent covariances
3. Correlation is the standardized covariance (covariance divided by the product of standard deviations)
4. Correlation values range from -1 to +1