(i) To find the probability that a randomly chosen athlete has a PB between 46 and 53 seconds, we standardize the values using the formula:

\(P(46 < X < 53) = P\left( \frac{46 - 49.2}{2.8} < Z < \frac{53 - 49.2}{2.8} \right)\)

This simplifies to:

\(P(-1.143 < Z < 1.357)\)

Using the standard normal distribution table, we find:

\(\Phi(1.357) + \Phi(1.143) - 1 = 0.9126 + 0.8735 - 1\)

Thus, the probability is:

0.786

(iii) To find the probability that exactly 2 out of 3 athletes have PBs of less than 46 seconds, we first find:

\(P(X < 46) = 0.1265\)

Then, using the binomial probability formula:

\(P(2 \text{ PB} < 46) = 3(1 - 0.1265)0.1265^2\)

This calculates to:

0.0419