(i) To find the number of sheep weighing between 70 kg and 72.5 kg, we first standardize the weights using the formula for the z-score:
\(z_1 = \frac{70 - 66.4}{5.6} = 0.6429\)
\(z_2 = \frac{72.5 - 66.4}{5.6} = 1.089\)
Next, we find the area under the standard normal curve between these z-scores:
\(\Phi(1.089) - \Phi(0.643) = 0.8620 - 0.7399 = 0.1221\)
Multiply this probability by the total number of sheep:
\(0.1221 \times 250 = 30.5\)
Therefore, approximately 30 or 31 sheep have a weight between 70 kg and 72.5 kg.
(ii) To find the value of \(y\), we use the symmetry of the normal distribution. The mean is 66.4 kg, and the proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more than \(y\) kg. Calculate the difference from the mean:
\(66.4 - 59.2 = 7.2\)
Thus, \(y = 66.4 + 7.2 = 73.6\) kg.