Understanding the Correlation Coefficient Calculation

The question asks us to calculate the correlation coefficient between two variables, x and y, given their covariance and variances.

The correlation coefficient, often denoted by $\rho$ (rho) or r, is a measure of the linear relationship between two variables. It 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.

Formula for Correlation Coefficient

The formula to calculate the population correlation coefficient ($\rho_{xy}$) between two variables x and y based on their covariance and standard deviations is:

$$ \rho_{xy} = \frac{\text{Cov}(x, y)}{\sigma_x \sigma_y} $$

Where:

• $\text{Cov}(x, y)$ is the covariance between x and y.
• $\sigma_x$ is the standard deviation of x.
• $\sigma_y$ is the standard deviation of y.

Remember that the standard deviation of a variable is the square root of its variance:

• $\sigma_x = \sqrt{\text{Variance}(x)}$
• $\sigma_y = \sqrt{\text{Variance}(y)}$

Applying the Given Data

We are provided with the following values:

• Covariance between x and y ($\text{Cov}(x, y)$) = 30
• Variance of x ($\text{Variance}(x)$) = 25
• Variance of y ($\text{Variance}(y)$) = 144

Step-by-Step Calculation

Let's calculate the standard deviations first:

Step 1: Calculate the standard deviation of x ($\sigma_x$)

$$ \sigma_x = \sqrt{\text{Variance}(x)} = \sqrt{25} $$

$$ \sigma_x = 5 $$

Step 2: Calculate the standard deviation of y ($\sigma_y$)

$$ \sigma_y = \sqrt{\text{Variance}(y)} = \sqrt{144} $$

$$ \sigma_y = 12 $$

Step 3: Substitute the values into the correlation coefficient formula

$$ \rho_{xy} = \frac{\text{Cov}(x, y)}{\sigma_x \sigma_y} $$

$$ \rho_{xy} = \frac{30}{5 \times 12} $$

$$ \rho_{xy} = \frac{30}{60} $$

Step 4: Calculate the final value

$$ \rho_{xy} = 0.5 $$

Thus, the correlation coefficient between x and y is 0.5.

Let's compare this result with the given options:

Our calculated value, 0.5, matches Option 2.

Revision Table: Key Concepts

Additional Information: Interpreting Correlation

The correlation coefficient provides insights into the relationship:

• A value close to +1 suggests a strong positive linear relationship (as x increases, y tends to increase proportionally).
• A value close to -1 suggests a strong negative linear relationship (as x increases, y tends to decrease proportionally).
• A value close to 0 suggests a weak or no linear relationship. Note that a correlation of 0 does not necessarily mean the variables are independent; they could have a non-linear relationship.
• The correlation coefficient is a dimensionless quantity, meaning it does not have any units. This makes it useful for comparing the strength of relationships between different pairs of variables.
• In this case, a correlation coefficient of 0.5 indicates a moderate positive linear relationship between x and y.