Questions: The following is a set of data from a sample of n=11 items. Complete parts (a) through (c). X 20 6 11 14 13 17 16 12 10 1 7 Y 40 12 22 28 26 34 32 24 20 2 14 a. Compute the sample covariance. (Round to three decimal places as needed.)

Solution Steps

The sample covariance between the 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, the calculated sample covariance is:

\[ \text{Cov}(X,Y) = 58.945 \]

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

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

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

The calculated standard deviations are:

\[ \sigma_X = 5.429 \]

\[ \sigma_Y = 10.858 \]

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

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

Substituting the values:

\[ r = \frac{58.945}{5.429 \times 10.858} = 1.0 \]

Final Answer