SQA Higher Applications of Mathematics Statistics and Probability: diagrams, correlation, hypothesis testing and expectation

What Statistics and Probability actually demands

Statistics is the largest single strand of the course and the basis of the project, which is a statistics report. The examiners test whether you can describe and compare data, measure relationships, make and judge inferences, and reason about uncertainty. Software produces the numbers; the marks are for choosing the right tool, interpreting the output, and concluding in context. This guide ties together the four dot-point pages of the module.

Statistical diagrams and sampling

The descriptive foundation: choose a diagram to suit the data (a box plot for comparing distributions, a histogram for grouped continuous data), and compare two data sets by one measure of centre (median or mean) and one of spread (interquartile range or standard deviation). An outlier lies beyond $Q_1 - 1.5 \times \text{IQR}$ or $Q_3 + 1.5 \times \text{IQR}$. Watch for misleading graphs such as a non-zero axis, and sample without bias using a method such as random, systematic or stratified sampling.

Correlation and regression

Pearson's correlation coefficient $r$ measures linear association from $-1$ to $+1$, stating strength and direction. The least-squares regression line $y = a + bx$ has slope $b$ (change in $y$ per unit of $x$) and intercept $a$ (the value of $y$ at $x = 0$), and predicts $y$ from $x$. Trust interpolation within the data and treat extrapolation beyond it with caution. A strong correlation never proves causation: allow for a confounding variable.

Hypothesis testing and confidence intervals

A hypothesis test weighs evidence against a null hypothesis $H_0$. Software returns a p-value, the probability of a result at least as extreme as observed if $H_0$ is true; reject $H_0$ when the p-value is below the significance level. A t-test is used when the population standard deviation is estimated from the sample (the usual case) and a z-test when it is known. A confidence interval gives a plausible range for a population value; a higher confidence level widens it and a larger sample narrows it.

Probability and expectation

A probability lies between $0$ and $1$. Combine events with the multiplication rule for "and" (independent events) and the addition rule for "or", and organise multi-stage events with tree diagrams, multiplying along paths and adding across them. Without replacement, later probabilities are conditional. The expected value $E = \sum (\text{value} \times \text{probability})$ is the long-run average outcome and the link to decision making under risk.

How Statistics and Probability is examined

A typical SQA profile for this area:

• Describing data. Choosing and reading diagrams, comparing centre and spread, flagging outliers and misleading graphs.
• Relationships. Interpreting $r$ and a regression line, predicting, and cautioning about causation.
• Inference. Stating hypotheses, interpreting a p-value and a confidence interval, choosing a t-test or z-test.
• Uncertainty. Combining probabilities, tree diagrams, and expected value.

Check your knowledge

A mix of recall and method questions covering the module. Attempt them, then check against the solutions.

1. A data set has $Q_1 = 20$ and $Q_3 = 32$. Find the IQR and the upper outlier boundary. (2 marks)
2. A correlation coefficient is $r = 0.92$. Describe the relationship. (1 mark)
3. A test gives a p-value of $0.03$ at the $5\%$ level. State the conclusion. (1 mark)
4. Interpret a $95\%$ confidence interval of $(7.2, 8.8)$ for a mean. (1 mark)
5. A prize pays $\pounds 20$ with probability $0.05$ and nothing otherwise. Find the expected value. (2 marks)