The Central Limit TheoremEdexcel A-Level Further Maths: Mind map

Exam questions on The Central Limit Theorem

1. The number of customers $X$ arriving at a small shop in an hour has a Poisson distribution with mean $4$. The manager records the number of arrivals in each of $50$ randomly chosen hours and calculates the sample mean $\bar{X}$.
   Find the probability that the sample mean lies between $3.8$ and $4.3$.2 marks
2. A box contains $12$ pens, each of which is faulty with probability $0.25$, independently of the others. Let $X$ be the number of faulty pens in a randomly chosen box. A random sample of $40$ boxes is taken and $\bar{X}$ is the mean number of faulty pens per box in the sample.
   Find $\mathrm{P}(\bar{X}>3.3)$.2 marks
3. At a fair, the number of tickets $X$ a visitor buys up to and including the first prize-winning ticket has a geometric distribution with parameter $p=0.2$. A random sample of $60$ visitors is taken, and $\bar{X}$ is the mean number of tickets bought up to and including the first prize.
   Explain why $\bar{X}$ can be assumed to have a normal distribution, and state its approximate distribution.3 marks