(a)(i) To find the probability that a randomly chosen member has a height less than 170 cm, we use the standard normal distribution formula:

\(P(X < 170) = P\left(Z < \frac{170 - 166}{10}\right) = P(Z < 0.4)\)

Using standard normal distribution tables, \(P(Z < 0.4) = 0.655\).

(a)(ii) We know that 40% of the members have heights greater than \(h\) cm, so:

\(P\left(Z > \frac{h - 166}{10}\right) = 0.4\)

This implies:

\(\frac{h - 166}{10} = 0.253\)

Solving for \(h\):

\(h = 166 + 10 \times 0.253 = 168.53\)

(b) Given \(\sigma = \frac{2}{3}\mu\), we need to find \(P(X > 0)\):

\(P(X > 0) = P\left(Z > \frac{0 - \mu}{\sigma}\right) = P\left(Z > \frac{0 - \mu}{\frac{2}{3}\mu}\right)\)

\(= P\left(Z > -\frac{3}{2}\right)\)

Using standard normal distribution tables, \(P(Z > -1.5) = 0.933\).