The Binomial Distribution and Sampling Distribution of the Mean

statisticalanalysis year12 ext1

The binomial distribution and sampling distribution of the mean | NSW Curriculum Website

• ME1-12-06 solves problems involving binomial distributions, sampling distribution of the mean and the central limit theorem

📖 Prior Knowledge

Content	Prerequisite relationships
Data Analysis	- Distinguish a census from a sample → Populations and samples
Probability	- Calculate expected frequency → Expected frequencies for a binomial
The Normal Distribution	- Calculate normal probabilities → Probabilities using the central limit theorem
Random Variables	- Define a discrete probability distribution → Define a Bernoulli random variable - Find the expected value of a discrete variable → Mean and variance of a Bernoulli variable - Find the expected value of a discrete variable → Sampling distribution of the mean - Identify features of the normal curve → The central limit theorem
Permutations and Combinations	- Count combinations using ${}^n{C}_r$ → Define a binomial random variable
The Binomial Theorem	- Expand using the binomial theorem → Use the binomial probability formula

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 $\mathrm{E}(X)=\mu =p$ to find the mean, and $\mathrm{Var}(X)=p(1-p)$ 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 $X\sim \mathrm{Bin}(n,p)$ 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 $\mathrm{E}(X)=\mu =np$ to find the mean, and $\mathrm{Var}(X)=np(1-p)$ to find the variance when $X\sim \mathrm{Bin}(n,p)$
• 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 $n\ge 30$
• 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 $n\ge 30$
• Define the sample mean for a sample $X_1,X_2,\dots ,X_n$ taken from a population as $\bar{X}=\frac{1}{n}\sum _{i=1}^n{X}_i$
• 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, $\bar{X}$, is a random variable that estimates the population mean, $\mu$, 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 $\mu$ and variance ${\sigma }^2$, provided the sample size $n$ is large enough $(n\ge 30)$, the sampling distribution is approximately normal with mean $\mu$ and variance $\frac{{\sigma }^2}{n}$; that is, $\bar{X}$ is approximately $N(\mu ,\frac{{\sigma }^2}{n})$
• 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 $E(\bar{X})=\mu$ and $\mathrm{Var}(\bar{X})=\frac{{\sigma }^2}{n}$ as the mean and the variance of the sampling distribution of the mean for samples of size $n$ drawn from a population with mean $\mu$ and variance ${\sigma }^2$
• 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