(i) To find the probability that a boy weighs more than 65 kg, we standardize the value:

\(P(X > 65) = P\left( Z > \frac{65 - 61.4}{12.3} \right) = P(Z > 0.2927)\)

Using the standard normal distribution table, \(P(Z > 0.2927) = 1 - 0.6153 = 0.385\).

(ii) We know \(P(X < 65) = 0.6153\), so \(P(65 < X < k) = 0.25\).

Thus, \(P(X < k) = 0.25 + 0.6153 = 0.8653\).

Find the z-value for 0.8653: \(z = 1.105\).

Standardize: \(1.105 = \frac{k - 61.4}{12.3}\).

Solve for \(k\): \(k = 1.105 \times 12.3 + 61.4 = 75.0\).

(iii) For Brigville, use the z-values for the given probabilities:

\(2.326 = \frac{97.2 - \mu}{\sigma}\) and \(-0.44 = \frac{55.2 - \mu}{\sigma}\).

Solve these simultaneous equations to find \(\mu\) and \(\sigma\):

From \(2.326\sigma = 97.2 - \mu\) and \(-0.44\sigma = 55.2 - \mu\),

Eliminate \(\mu\) to find \(\sigma = 15.2\).

Substitute back to find \(\mu = 61.9\).