2 A particular genetic mutation occurs in one in every 300 births on average. A random sample of 1200 births is selected.
- State the exact distribution of \(X\), the number of births in the sample which have the mutation.
- Explain why \(X\) has, approximately, a Poisson distribution.
- Use a Poisson approximating distribution to find
(A) \(\mathrm { P } ( X = 1 )\),
(B) \(\mathrm { P } ( X > 4 )\). - Twenty independent samples, each of 1200 births, are selected. State the mean and variance of a Normal approximating distribution suitable for modelling the total number of births with the mutation in the twenty samples.
- Use this Normal approximating distribution to
(A) find the probability that there are at least 90 births which have the mutation,
( \(B\) ) find the least value of \(k\) such that the probability that there are at most \(k\) births with this mutation is greater than 5\%.