Course

# VCE Specialist Mathematics Units 3 and 4 – Probability and Statistics

6.1 Expectation
6.2 Sample Mean
6.3 Discrete Random Variables
6.4 Continuous Random Variables
6.5 Hypothesis Testing

0 Lessons

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