3 UPVC facia board is supplied in lengths labelled as 5 metres. The actual length, \(X\) metres, of a board may be modelled by a normal distribution with a mean of 5.08 and a standard deviation of 0.05 .
- Determine:
- \(\mathrm { P } ( X < 5 )\);
- \(\mathrm { P } ( 5 < X < 5.10 )\).
- Determine the probability that the mean length of a random sample of 4 boards:
- exceeds 5.05 metres;
- is exactly 5 metres.
- Assuming that the value of the standard deviation remains unchanged, determine the mean length necessary to ensure that only 1 per cent of boards have lengths less than 5 metres.