(i) To find the proportion of gem stones in each category, we standardize the weights using the formula:

\(z = \frac{x - \mu}{\sigma}\)

For Small (\(X < 1.2\)): \(z = \frac{1.2 - 1.9}{0.55} = -1.2727\)

\(P(X < 1.2) = P(z < -1.2727) = 0.1014\)

For Large (\(X > 2.5\)): \(z = \frac{2.5 - 1.9}{0.55} = 1.0909\)

\(P(X > 2.5) = P(z > 1.0909) = 0.138\)

For Medium (\(1.2 < X < 2.5\)): \(P(1.2 < X < 2.5) = 1 - P(X < 1.2) - P(X > 2.5) = 1 - 0.101 - 0.138 = 0.761\)

(ii) We need \(P(k < X < 2.5) = 0.8\). We know \(P(X < 2.5) = 0.8623\), so \(P(X > k) = 0.9377\).

Find \(z\) such that \(P(z) = 0.9377\), which gives \(z = -1.536\).

Using the standardization formula:

\(-1.536 = \frac{k - 1.9}{0.55}\)

Solving for \(k\):

\(k = 1.9 + (-1.536 \times 0.55) = 1.06\)