Questions: The correlation coefficient is always a number between -1 and 1.

Solution Steps

The covariance between two variables \( X \) and \( Y \) is calculated using the formula:

\[ \text{Cov}(X,Y) = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y}) \]

For the given data, we find:

\[ \text{Cov}(X,Y) = 5.0 \]

The standard deviation for \( X \) and \( Y \) is calculated using the formulas:

\[ \sigma_X = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2} \]

\[ \sigma_Y = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (Y_i - \bar{Y})^2} \]

For the given data, we find:

\[ \sigma_X = 1.58 \] \[ \sigma_Y = 3.16 \]

The correlation coefficient \( r \) is calculated using the formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

Substituting the values we calculated:

\[ r = \frac{5.0}{1.58 \times 3.16} = 1.0 \]

Final Answer