1 The random variable \(X\) represents the time taken in minutes for a haircut at a barber's shop. \(X\) is Normally distributed with mean 11 and standard deviation 3 .

1. Find \(\mathrm { P } ( X < 10 )\).
2. Find the probability that exactly 3 out of 8 randomly selected haircuts take less than 10 minutes.
3. Use a suitable approximating distribution to find the probability that at least 50 out of 100 randomly selected haircuts take less than 10 minutes. A new hairdresser joins the shop. The shop manager suspects that she takes longer on average than the other staff to do a haircut. In order to test this, the manager records the time taken for 25 randomly selected cuts by the new hairdresser. The mean time for these cuts is 12.34 minutes. You should assume that the time taken by the new hairdresser is Normally distributed with standard deviation 3 minutes.
4. Write down suitable null and alternative hypotheses for the test.
5. Carry out the test at the \(5 \%\) level.