Last updated on Oct 23, 2026

Central Limit Theorem (Edexcel A-Level Further Mathematics): Quizzes

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Central Limit Theorem

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When is $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$ exact rather than approximate?

When the population is normal

For non-normal populations, when does the CLT give an approximate distribution for $\bar{X}$ ?

When sample size $n$ is large

What does the CLT state about $\bar{X}$ for any distribution with mean $\mu$ , variance $\sigma^2$ ?

$\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$

What sample size threshold is mentioned as sufficient for applying the CLT?

$n \geq 25$

When applying the CLT, how is the population variance $\sigma^2$ typically estimated?

By sample variance $s^2$

For a discrete variable $Y$ , how is $E(Y)$ calculated from its probability distribution?

$\sum y \cdot P(Y=y)$

What is the formula for $\text{Var}(Y)$ in terms of $E(Y^2)$ and $E(Y)$ ?

$E(Y^2) - [E(Y)]^2$

For $n=50$ observations, by what increment do possible values of $\bar{Y}$ increase?

$\frac{1}{50}$ or $0.02$

What is the formula for sample variance $s^2$ from $\bar{x^2}$ and $\bar{x}$ ?

$s^2 = \frac{n}{n-1}(\bar{x^2}-\bar{x}^2)$

If p-value $= 0.01786$ and significance level $= 0.001$ , what is the decision?

Accept $H_0$

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