11–12Mathematics Extension 1 11–12 Syllabus

Record of changes

Implementation from 2026

Expand for detailed implementation advice

Overview Rationale Aim Outcomes Content Assessment Glossary Teaching and learning support

Content

Year 11

Year 12

Teaching advice

Examples

The binomial distribution and sampling distribution of the mean

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
ME1-12-06
solves problems involving binomial distributions, sampling distribution of the mean and the central limit theorem

The binomial distribution and sampling distribution of the mean

Bernoulli distributions

Define a Bernoulli random variable as a model for two-outcome situations, referred to as success and failure
Model a Bernoulli trial using a random variable, $X$ , where the probability of success, $X = 1$ , is $p$ and the probability of failure, $X = 0$ , is $q = 1 - p$
Define a Bernoulli distribution as the discrete probability distribution of a Bernoulli random variable
Establish and use the formulae $E (X) = μ = p$ to find the mean, and $Var (X) = p (1 - p)$ to find the variance of the Bernoulli distribution with parameter $p$
Identify contexts suitable to be modelled by Bernoulli random variables
Solve practical problems involving Bernoulli random variables

Binomial distributions

Define a binomial random variable, $X$ , as the number of ‘successes’ in $n$ independent and identically distributed Bernoulli trials, with the same probability of success $p$ in each trial, where $n$ is a fixed finite number
Identify contexts suitable to be modelled using binomial distributions
Define a binomial experiment as a fixed number of Bernoulli trials
Use the notation $X ~ Bin (n, p)$ to represent a binomial random variable, $X$ , where $n$ is the number of trials and probability of success is $p$
Recognise the significance of $^{n} C_{r}$ as the number of ways in which an outcome with $r$ successes can occur in $n$ trials
Use the formula $P (X = r) =^{n} C_{r} p^{r} {(1 - p)}^{n - r}$ to find the probability of $r$ successes in a binomial experiment involving $n$ trials
Calculate the expected frequencies of the various possible outcomes from a binomial experiment, where the expected frequency of an outcome happening $r$ times in $n$ trials is $n \times P (X = r)$
Use the formulae $E (X) = μ = np$ to find the mean, and $Var (X) = np (1 - p)$ to find the variance when $X ~ Bin (n, p)$
Solve practical problems involving binomial distributions and binomial probabilities with and without online computational applications, excluding the normal approximation to the binomial distribution

Sampling distribution of the mean and the central limit theorem

Define a statistical population as the entire group of people or objects about which information is sought
Define a sample as a selection of people or objects drawn from a population
Recognise that a statistic taken from a sample will be a good approximation if the sample is random and the sample size is large enough, typically $n \geq 30$
Recognise that, in general, the distribution of a population and a statistic, such as its mean, are unknown
Recognise that it is likely that the distribution of a random sample will resemble the distribution of the population when the sample size, $n$ , is large enough, typically $n \geq 30$
Define the sample mean for a sample $X_{1}, X_{2}, \dots, X_{n}$ taken from a population as $\overset{̅}{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}$
Recognise that the sample means obtained from repeated sampling may be different, even if all the samples are of the same size
Define, for a given sample size $n$ , the sampling distribution of the mean to be the distribution of the sample means of all samples of size $n$
Recognise that sample mean, $\overset{̅}{X}$ , is a random variable that estimates the population mean, $μ$ , when the size of the sample is large enough and consider its usefulness in making predictions in a variety of contexts including politics, finance, agriculture and biology
State the central limit theorem as: for a population with mean $μ$ and variance $σ^{2}$ , provided the sample size $n$ is large enough ( $n \geq 30$ ), the sampling distribution is approximately normal with mean $μ$ and variance $\frac{σ^{2}}{n}$ ; that is, $\overset{̅}{X}$ is approximately $N (μ, \frac{σ^{2}}{n})$
Recognise the significance of the central limit theorem for populations that are not necessarily normally distributed; that, irrespective of the population distribution, for a sufficiently large sample size, the sampling distribution of the mean is approximately normal
Use the formulas $E (\overset{̅}{X}) = μ$ and $Var (\overset{̅}{X}) = \frac{σ^{2}}{n}$ as the mean and the variance of the sampling distribution of the mean for samples of size $n$ drawn from a population with mean $μ$ and variance $σ^{2}$
Examine the effect of the sample size on the variance of sample means with digital tools
Apply the central limit theorem to estimate the probability that the sample mean lies within given bounds

Related files

Welcome to the NSW Curriculum website

11–12Mathematics Extension 1 11–12 Syllabus

Content