Central Limit Theorem

Section FS1-5

1 spec leaf

Notes and three levels of exam-style practice for each registered specification leaf in this section.

Open the printable pack

FS1-5.1

Applications of the Central Limit Theorem to other distributions.

For a large independent random sample from a population with mean $\mu$ and variance $\sigma^2$ , $\overline X\mathrel{\dot\sim}N(\mu,\sigma^2/n)$ .
Standardise a sample mean with standard error $\sigma/\sqrt n$ : $Z=(\overline X-\mu)/(\sigma/\sqrt n)$ .
The Central Limit Theorem can be applied to populations from the A-level Mathematics and FS₁ distributions when the sample is sufficiently large.
A common error is to use population variance $\sigma^2$ for $\overline X$ instead of dividing it by $n$ .

Tier 1 · Easy

4 marks

ORIGINAL

Independent lifetimes have population mean $50$ hours and variance $100$ hours $^2$ . For a sample of $36$ lifetimes, use the Central Limit Theorem to estimate $P(\overline X<47.5)$ .

Tier 2 · Standard

6 marks

ORIGINAL

A geometric population has parameter $p=0.25$ . A random sample of $64$ observations is taken. Use the Central Limit Theorem to estimate $P(3.5<\overline X<4.4)$ .

Tier 3 · Hard

6 marks

ORIGINAL

A population has mean $\mu$ and variance $36$ . Using the Central Limit Theorem normal approximation, estimate the least sample size $n$ for which $P(|\overline X-\mu|<1.2)\geq0.95$ .