Does the CLT need the population to be normal?

No. It applies to any population with a mean and variance.

Give \mu and \sigma^2 for Po(\lambda).

\mu=\lambda and \sigma^2=\lambda

Give \mu and \sigma^2 for Geo(p).

\mu=\frac1p and \sigma^2=\frac{1-p}{p^2}

Give \mu and \sigma^2 for NB(r,p).

\mu=\frac rp and \sigma^2=\frac{r(1-p)}{p^2}

How do you standardise a sample mean?

z=\frac{\bar{x}-\mu}{\sigma/\sqrt n}

How does P(|\bar{X}-\mu|<a) change as n increases?

It increases, because the standard deviation \frac{\sigma}{\sqrt n} decreases.

How do you justify using the CLT in an exam answer?

The sample size is large, so the sample mean is approximately normal even though the population is not.

The Central Limit TheoremEdexcel A-Level Further Maths: Flashcards

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State the Central Limit Theorem.

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State the Central Limit Theorem.: For a population with mean $\mu$ and variance $\sigma^2$ , for large $n$ the sample mean is approximately $\mathrm{N}\left(\mu,\frac{\sigma^2}{n}\right)$ .
Does the CLT need the population to be normal?: No. It applies to any population with a mean and variance.
What is the standard deviation of the sample mean?: $\frac{\sigma}{\sqrt n}$
What is the variance of the sample mean?: $\frac{\sigma^2}{n}$
What sample size is usually considered large?: About $30$ or more.
Give $\mu$ and $\sigma^2$ for $\mathrm{Po}(\lambda)$ .: $\mu=\lambda$ and $\sigma^2=\lambda$
Give $\mu$ and $\sigma^2$ for $\mathrm{B}(n,p)$ .: $\mu=np$ and $\sigma^2=np(1-p)$
Give $\mu$ and $\sigma^2$ for $\mathrm{Geo}(p)$ .: $\mu=\frac1p$ and $\sigma^2=\frac{1-p}{p^2}$
Give $\mu$ and $\sigma^2$ for $\mathrm{NB}(r,p)$ .: $\mu=\frac rp$ and $\sigma^2=\frac{r(1-p)}{p^2}$
How do you standardise a sample mean?: $z=\frac{\bar{x}-\mu}{\sigma/\sqrt n}$
$X\sim\mathrm{Po}(4)$ , $n=50$ . What is the distribution of $\bar{X}$ ?: $\mathrm{N}(4,0.08)$
How does $\mathrm{P}(|\bar{X}-\mu|<a)$ change as $n$ increases?: It increases, because the standard deviation $\frac{\sigma}{\sqrt n}$ decreases.
How do you justify using the CLT in an exam answer?: The sample size is large, so the sample mean is approximately normal even though the population is not.

Exam questions on The Central Limit Theorem

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
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
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

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).