To find the probability that a randomly chosen salmon is 34 cm long, we need to consider the range from 33.5 cm to 34.5 cm due to rounding to the nearest centimetre.

We standardize the values using the formula for the standard normal distribution:

\(Z = \frac{X - \mu}{\sigma}\)

For 34.5 cm: \(Z = \frac{34.5 - 32.9}{2.4} = 0.667\)

For 33.5 cm: \(Z = \frac{33.5 - 32.9}{2.4} = 0.25\)

We find the probabilities using the standard normal distribution table:

\(\Phi(0.667) = 0.7477\)

\(\Phi(0.25) = 0.5987\)

The probability that a salmon is 34 cm long is the difference between these probabilities:

\(0.7477 - 0.5987 = 0.149\)