(i) Let \(n\) be the number of children. The mean height is given by:

\(\frac{\Sigma x}{n} = 104.8\)

We know \(\Sigma(x - 100) = 72\), so:

\(\Sigma x - 100n = 72\)

Substitute \(\Sigma x = 104.8n\) into the equation:

\(104.8n - 100n = 72\)

\(4.8n = 72\)

\(n = \frac{72}{4.8} = 15\)

(ii) We need to find \(\Sigma(x - 104.8)^2\). We have:

\(\Sigma(x - 100)^2 = 499.2\)

\(\Sigma(x - 104.8)^2 = \Sigma(x - 100 + 100 - 104.8)^2\)

\(= \Sigma((x - 100) - 4.8)^2\)

Using the expansion \((a - b)^2 = a^2 - 2ab + b^2\), we have:

\(\Sigma(x - 104.8)^2 = \Sigma(x - 100)^2 - 2 \times 4.8 \times \Sigma(x - 100) + 15 \times 4.8^2\)

\(= 499.2 - 2 \times 4.8 \times 72 + 15 \times 4.8^2\)

\(= 499.2 - 691.2 + 345.6\)

\(= 153.6\)