1 In large-scale tree-felling operations, a machine cuts down trees, strips off the branches and then cuts the trunks into logs of length \(X\) metres for transporting to a sawmill. It may be assumed that values of \(X\) are normally distributed with mean \(\mu\) and standard deviation 0.16 , where \(\mu\) can be set to a specific value.

1. Given that \(\mu\) is set to 3.3 , determine:
   1. \(\mathrm { P } ( X < 3.5 )\);
   2. \(\mathrm { P } ( X > 3.0 )\);
   3. \(\mathrm { P } ( 3.0 < X < 3.5 )\).
2. The sawmill now requires a batch of logs such that there is a probability of 0.025 that any given log will have a length less than 3.1 metres. Determine, to two decimal places, the new value of \(\mu\).