Statistical inference is the process of using data from a sample to make estimates or draw conclusions about an entire population. In VCE Specialist Mathematics, this focuses on estimating the population mean ($\mu$) and the population proportion ($p$).
KEY TAKEAWAY: A point estimate provides no information about the precision of the estimate; a confidence interval provides a range of plausible values and a level of certainty.
A confidence interval for the population mean is constructed when the population standard deviation ($\sigma$) is known, or when the sample size ($n$) is large enough ($n \ge 30$) to use the sample standard deviation ($s$) as an approximation for $\sigma$.
The $C\%$ confidence interval for $\mu$ is given by:
$$\bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}}$$
Where:
* $\bar{x}$ is the sample mean.
* $z$ is the critical value for the desired confidence level.
* $\frac{\sigma}{\sqrt{n}}$ is the standard error of the mean.
* $M = z \cdot \frac{\sigma}{\sqrt{n}}$ is the margin of error.
The value of $z$ is determined by the standard normal distribution $Z \sim N(0, 1)$.
| Confidence Level | $z$ value (approx) | $z$ value (exact) |
|---|---|---|
| 90% | 1.645 | $invNorm(0.95, 0, 1)$ |
| 95% | 1.96 | $invNorm(0.975, 0, 1)$ |
| 99% | 2.576 | $invNorm(0.995, 0, 1)$ |
Note: For some technology-free questions, VCAA may specify using $z \approx 2$ for a 95% confidence interval.
EXAM TIP: If a question asks for the “distance between the sample mean and the population mean,” they are asking for the Margin of Error ($M$).
The width ($W$) of a confidence interval is the distance between the upper and lower bounds.
$$W = \text{Upper Bound} - \text{Lower Bound} = 2 \times \text{Margin of Error}$$
$$W = 2z \frac{\sigma}{\sqrt{n}}$$
To find the minimum sample size $n$ required to achieve a specific margin of error $M$:
$$n = \left( \frac{z \cdot \sigma}{M} \right)^2$$
Always round $n$ up to the nearest whole number to ensure the margin of error is not exceeded.
VCAA FOCUS: Questions often ask how the sample size must change to achieve a certain reduction in width. If the width is reduced by a factor of $k$, the sample size must increase by a factor of $\frac{1}{k^2}$. For example, to reduce width by $\frac{2}{3}$ (to $\frac{1}{3}$ of the original), $n$ must increase by $3^2 = 9$.
When dealing with categorical data (e.g., “Yes/No” responses), we estimate the population proportion $p$ using the sample proportion $\hat{p} = \frac{x}{n}$.
For a large sample size, the distribution of $\hat{P}$ is approximately normal: $\hat{P} \approx N\left(p, \frac{p(1-p)}{n}\right)$.
The $C\%$ confidence interval for $p$ is:
$$\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$
This approximation is generally considered valid if $n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$.
COMMON MISTAKE: Using the population proportion $p$ in the standard error formula when constructing a confidence interval. Since $p$ is unknown (that’s why we are making the interval!), we must use the sample estimate $\hat{p}$ to calculate the standard error.
It is a common misconception that a 95% confidence interval has a “95% probability of containing the population mean.”
| Feature | Population Mean ($\mu$) | Population Proportion ($p$) |
|---|---|---|
| Point Estimate | $\bar{x}$ | $\hat{p} = \frac{x}{n}$ |
| Standard Error | $\frac{\sigma}{\sqrt{n}}$ | $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ |
| Confidence Interval | $\bar{x} \pm z\frac{\sigma}{\sqrt{n}}$ | $\hat{p} \pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ |
STUDY HINT: Practice using your CAS calculator to find confidence intervals quickly. In the Statistics menu, look for
One-Sample Z Intervalfor means andOne-Prop Z Intervalfor proportions. Knowing how to do this manually is essential for “Show that” or technology-free questions.
