2 Suppose that 3\% of the population of a large city have red hair.
- A random sample of 10 people from the city is selected. Find the probability that there is at least one person with red hair in this sample.
A random sample of 60 people from the city is selected. The random variable \(X\) represents the number of people in this sample who have red hair.
- Explain why the distribution of \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
- Hence find
(A) \(\mathrm { P } ( X = 2 )\),
(B) \(\mathrm { P } ( X > 2 )\). - Discuss whether or not it would be appropriate to model \(X\) using a Normal approximating distribution.
A random sample of 5000 people from the city is selected.
- State the exact distribution of the number of people with red hair in the sample.
- Use a suitable Normal approximating distribution to find the probability that there are at least 160 people with red hair in the sample.