Questions: Lengths and Girths Length x 158 159 159 158 159 158 157 156 155 Girth y 125 123 121 120 118 117 115 110 109

Solution Steps

To calculate the covariance \( \text{Cov}(X,Y) \) and the correlation coefficient \( r \) between the lengths \( X \) and girths \( Y \):

\[ \text{Cov}(X,Y) = 6.58 \]

The standard deviations are calculated as follows:

\[ \sigma_X = 1.41 \] \[ \sigma_Y = 5.48 \]

The correlation coefficient \( r \) is given by the formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} = \frac{6.58}{1.41 \times 5.48} \approx 0.85 \]

The mean \( \mu \) of the lengths \( X \) is calculated as:

\[ \mu_X = \frac{\sum_{i=1}^N x_i}{N} = \frac{1419}{9} \approx 157.67 \]

The mean \( \mu \) of the girths \( Y \) is calculated as:

\[ \mu_Y = \frac{\sum_{i=1}^N y_i}{N} = \frac{1058}{9} \approx 117.56 \]

The variance \( \sigma^2 \) and standard deviation \( \sigma \) for the lengths \( X \) are calculated as follows:

\[ \sigma^2_X = \frac{\sum (x_i - \mu_X)^2}{n-1} = 2.0 \] \[ \sigma_X = \sqrt{2.0} \approx 1.41 \]

For the girths \( Y \):

\[ \sigma^2_Y = \frac{\sum (y_i - \mu_Y)^2}{n-1} = 30.03 \] \[ \sigma_Y = \sqrt{30.03} \approx 5.48 \]

Final Answer