In this area of study students cover statistical inference related to the definition and distribution of sample means, simulations and confidence interval.

Linear combinations of random variables, including:

- for random variables \(X\) and \(Y\), \( E(aX + b) = aE(X) + b\) and \(E(aX + bY) = aE(X) + b E(Y) \)
- for random variables \(X\) and \(Y\), \( \text{Var}(aX +b) = a^2 \text{Var}(X) \) and for independent random variables \( X \) and \( Y \), \( \text{Var}(aX +bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) \)
- for independent random variables \(X\) and \(Y\) with normal distributions then \(aX + bY\) also has a normal distribution.

Sample means, including:

- concept of the sample mean \( \overline{X} \) as a random variable whose value varies between samples where \(X\) is a random variable with mean \( \mu \) and standard deviation \( σ \)
- simulation of repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate properties of the distribution of \( \overline{X} \) across samples of a fixed size \(n\) including its mean \( \mu \) and its standard deviation \( \displaystyle \frac{\sigma}{\sqrt{n}} \), where \( \mu \) and \( \sigma \) are the mean and standard deviation of \( X \), and its approximate normality if \( n \) is large.

Confidence intervals for means, including:

- determination of confidence intervals for means and the use of simulation to illustrate variations in confidence intervals between samples and to show that most but not all confidence intervals contain \( \mu \)
- construction of an approximate confidence interval \( \displaystyle \left( \overline{X}-z\frac{s}{\sqrt{n}}, \overline{X}+z\frac{s}{\sqrt{n}} \right) \) where \(s\) is the sample standard deviation and \( z \) is the appropriate quantile for the standard normal distribution, in particular the \( 95\% \) confidence interval as an example of such an interval where \( z \approx 1.96 \) (the term standard error may be used but is not required).

Hypothesis testing for a population mean with a sample drawn from a normal distribution of known variance or for a large sample, including:

- \(p\) values for hypothesis testing related to the mean
- formulation of a null hypothesis and an alternative hypothesis
- errors in hypothesis testing

*source – VCE Mathematics Study Design*