NSW Syllabuses

# Mathematics Extension 1 Stage 6 - Year 12 Extension 1 - Statistical Analysis ME-S1 The Binomial Distribution

## Outcomes

#### A student:

• ME12-5

applies appropriate statistical processes to present, analyse and interpret data

• ME12-6

chooses and uses appropriate technology to solve problems in a range of contexts

• ME12-7

evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms

## Subtopic Focus

The principal focus of this subtopic is to develop an understanding of binomial random variables and their uses in modelling random processes involving chance and variation.

Students develop understanding and appreciation of binomial distributions and associated statistical analysis methods and their use in modelling binomial events.

## Content

• S1.1: Bernoulli and binomial distributions
• Students:
• use a Bernoulli random variable as a model for two-outcome situations and use Bernoulli random variables and their associated probabilities to solve practical problems (ACMMM143, ACMMM146)
• identify contexts suitable for modelling by Bernoulli random variables (ACMMM144)
• understand and apply the formulae for the mean, $$\bar x = p$$, and variance, $$p(1-p)$$, of the Bernoulli distribution with parameter $$p$$ (ACMMM145)
• understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in $$n$$ independent Bernoulli trials, with the same probability of success $$p$$ in each trial (ACMMM147)
• calculate the expected frequencies of the various possible outcomes from a series of Bernoulli trials
• identify contexts suitable for modelling by binomial random variables (ACMMM148)
• identify the binomial parameter $$p$$ as the probability of success
• apply the formulae for probabilities $$P(X=r) ={^n}C_r \times p^r(1-p)^{n-r}$$ associated with the binomial distribution with parameters $$n$$ and $$p$$ and understand the meaning of $$^n C_r$$ as the number of ways in which an outcome can occur (ACMMM149)
• understand and apply the formulae for the mean, $$\bar x = np$$, and the variance, $$np(1-p)$$, of a binomial distribution with parameters $$n$$ and $$p$$ (ACMMM149)
• S1.2: Normal approximation for the sample proportion
• Students:
• use appropriate graphs to explore the behaviour of the sample proportion on collected or supplied data
• understand the concept of the sample proportion $$\hat p$$ as a random variable whose value varies between samples (ACMMM174)
• explore the behaviour of the sample proportion using simulated data
• examine the approximate normality of the distribution of $$\widehat{p}$$ for large samples (ACMMM175)
• understand and use the normal approximation to the distribution of the sample proportion and its limitations