(i) Given that 80% of the days, Rafa spends more than 1.35 hours, we have:

\(P(X > 1.35) = 0.8\)

Using the standard normal distribution, \(z = -0.842\) for 20% in the left tail:

\(P\left( \frac{1.35 - 1.9}{\sigma} < z \right) = 0.2\)

\(-0.842 = \frac{1.35 - 1.9}{\sigma}\)

\(\sigma = \frac{-0.55}{-0.842} = 0.653\)

(ii) To find \(P(X < 2)\):

\(P\left( z < \frac{2 - 1.9}{0.6532} \right) = P(z < 0.1531)\)

Using the standard normal table, \(P(z < 0.1531) = 0.561\).

(iii) For a sample of 200 days, \(X \sim N(160, 32)\):

We need \(P(162.5 < X < 173.5)\):

\(P\left( \frac{162.5 - 160}{\sqrt{32}} < z < \frac{173.5 - 160}{\sqrt{32}} \right)\)

\(P(0.442 < z < 2.386) = \Phi(2.386) - \Phi(0.442)\)

\(= 0.9915 - 0.6707 = 0.321\)