Statistical Inference: study guide to the SQA Advanced Higher Statistics second area

Statistical Inference is the second of the three areas of SQA Advanced Higher Statistics and the bridge between describing data and testing claims about it. It teaches you to use a sample to estimate an unknown population value and to quantify how uncertain that estimate is. This guide maps the area and links to the full topic pages.

What the area covers

The area moves from how data is sampled, through how the sample mean behaves, to how a population value is estimated.

• Sampling methods. Simple random, systematic and stratified sampling, the difference between a population and a sample and between a parameter and a statistic, and how a poor method introduces bias.
• Sampling distributions and the central limit theorem. The sampling distribution of the sample mean, its standard error $\dfrac{\sigma}{\sqrt{n}}$, and the central limit theorem that makes the sample mean approximately normal.
• Estimation and confidence intervals. Point estimates of a mean and variance, confidence intervals for a mean using the normal and Student's t-distributions, and a confidence interval for a proportion.
• The statistical investigation. Posing a question, planning and collecting data, selecting and applying analysis, and communicating justified conclusions with limitations.

How the topics connect

The area has a clear logic. Sampling methods decide whether the sample is fit to represent the population at all. The central limit theorem then describes how the sample mean varies around the true mean, supplying the standard error that every interval and test depends on. Estimation uses that standard error to turn a single sample mean into a confidence interval, choosing the normal or t-distribution according to what is known about $\sigma$. The statistical investigation finally combines all of this with the design ideas from the first area and the tests of the third into one coherent process. The standard error and the choice between normal and t carry straight over into the hypothesis-testing area.

How to study this area

1. Memorise the standard error. Almost everything here uses $\dfrac{\sigma}{\sqrt{n}}$ (or $\dfrac{s}{\sqrt{n}}$); make it automatic and never drop the $\sqrt{n}$.
2. Be able to state the central limit theorem precisely. Include the mean, the variance $\dfrac{\sigma^2}{n}$ and the key point that the parent population can be any shape.
3. Learn the normal-versus-t decision. Known $\sigma$ or large $n$ means $z$; unknown $\sigma$ with small $n$ means t with $n - 1$ degrees of freedom.
4. Interpret intervals correctly. Practise stating the long-run-frequency meaning of a confidence level, a common written mark.
5. Rehearse choosing a method. For the investigation, drill matching the analysis (interval, t-test, non-parametric test, chi-squared, regression) to the question and data type.

Where to go next

Work through the four topic pages from this area, then test yourself with the area quiz. After that, move on to the Hypothesis Testing area, which uses the standard error and sampling distributions developed here to test claims about populations.