The Central Limit TheoremEdexcel A-Level Further Maths: Subtopic test

1. 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 $\bar{X}$.

   (a)

   Which of the following is the approximate distribution of $\bar{X}$?

   [1 mark]

   • A$\mathrm{N}(4,\,4)$
   • B$\mathrm{N}(4,\,0.08)$
   • C$\mathrm{N}(4,\,0.28)$
   • D$\mathrm{N}(200,\,200)$

   (b)

   Find $\mathrm{P}(\bar{X}>4.5)$.

   [1 mark]

   • A$0.0385$
   • B$0.0771$
   • C$0.4013$
   • D$0.9615$

   (c)

   Find the probability that the sample mean lies between $3.8$ and $4.3$.

   [2 marks]

   Total for question 1: 4 marks

2. 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.

   (a)

   What is the approximate variance of $\bar{X}$?

   [1 mark]

   • A$0.2372$
   • B$2.25$
   • C$0.0014$
   • D$0.0563$

   (b)

   Find $\mathrm{P}(\bar{X}<2.8)$.

   [1 mark]

   • A$0.4470$
   • B$0.8005$
   • C$0.1995$
   • D$0.0002$

   (c)

   Find $\mathrm{P}(\bar{X}>3.3)$.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   Explain why $\bar{X}$ can be assumed to have a normal distribution, and state its approximate distribution.

   [3 marks]

   (b)

   Find the probability that the sample mean is within $0.4$ of the population mean, that is $\mathrm{P}(4.6<\bar{X}<5.4)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A telesales agent needs $X$ calls to make $3$ sales, where $X$ has a negative binomial distribution with $r=3$ and $p=0.25$, independently for each target. In a period she completes $36$ targets, and $\bar{X}$ is the mean number of calls per target in the period.

   (a)

   (i) Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.
   (ii) Use the Central Limit Theorem to find the probability that her mean number of calls per target exceeds $13$.

   [6 marks]

   (b)

   (i) Find the probability that her mean number of calls per target in the $36$ targets is less than $11.2$.
   (ii) The agent completes $n$ targets. Find the smallest value of $n$ for which $\mathrm{P}(\bar{X}>13)<0.01$.

   [6 marks]

   Total for question 4: 12 marks

End of questions