AQA A-Level Further Mathematics Further statistics: a complete overview of discrete variables, Poisson, testing and confidence intervals

What Further statistics actually demands

Further statistics is one of the optional applied areas of AQA A-Level Further Mathematics. It extends the statistics of A-Level Mathematics with new distributions and tests, and rewards disciplined method: a clear null and alternative hypothesis, the right critical value or degrees of freedom, and a conclusion stated in context. This guide walks through all five topics, then sets out the exam patterns AQA repeats.

Discrete random variables

Discrete random variables are described by a probability distribution whose probabilities sum to $1$. The expectation is $E(X) = \sum x\, P(X = x)$ and the variance is $\operatorname{Var}(X) = E(X^2) - [E(X)]^2$. Under linear coding, $E(aX + b) = aE(X) + b$ and $\operatorname{Var}(aX + b) = a^2\operatorname{Var}(X)$, so an additive constant shifts the mean but not the variance. The same machinery gives the expectation of functions such as $X^2$.

The Poisson distribution

The Poisson distribution models random events occurring singly, independently and at a constant rate $\lambda$ in a fixed interval, with $P(X = x) = e^{-\lambda}\frac{\lambda^x}{x!}$ and mean equal to variance equal to $\lambda$. Independent Poisson variables add, $\text{Po}(\lambda_1) + \text{Po}(\lambda_2) = \text{Po}(\lambda_1 + \lambda_2)$, and the Poisson approximates a binomial $\text{B}(n, p)$ when $n$ is large and $p$ small, using $\lambda = np$.

Further hypothesis testing

Further hypothesis testing covers tests for a Poisson mean using Poisson probabilities and tests for a population mean using the normal statistic $Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$. You choose a one-tailed or two-tailed test from the wording, compare a probability or statistic with the significance level or critical value, and conclude in context. A Type I error rejects a true null; a Type II error keeps a false null.

Chi-squared tests

Chi-squared tests use the statistic $\chi^2 = \sum \frac{(O - E)^2}{E}$ to compare observed and expected frequencies. Goodness of fit tests whether data follow a stated distribution, pooling cells with expected frequency below $5$ and subtracting one degree of freedom per estimated parameter. Contingency tables test independence with $(\text{rows} - 1)(\text{columns} - 1)$ degrees of freedom, and a $2 \times 2$ table uses Yates' continuity correction.

Confidence intervals

Confidence intervals give a range of plausible values for a population mean. With known variance, a $C\%$ interval is $\bar{x} \pm z\frac{\sigma}{\sqrt{n}}$, where $\frac{\sigma}{\sqrt{n}}$ is the standard error. A higher confidence level widens the interval and a larger sample narrows it; the confidence level is the long-run proportion of intervals capturing the true mean. With unknown variance, the t distribution replaces $z$.

How Further statistics is examined

A typical AQA profile for Further statistics:

• Distribution calculations. Mean and variance of a discrete variable, Poisson probabilities, and combining or approximating distributions.
• Hypothesis tests. Stating hypotheses, choosing the tail, computing the probability or statistic, and concluding in context, including identifying error types.
• Chi-squared and intervals. Goodness of fit and independence tests with correct degrees of freedom, and confidence intervals with correct interpretation.

Check your knowledge

A mix of calculation and interpretation questions across Further statistics. Attempt them under timed conditions, then check against the solutions.

1. A variable takes values $0$ and $2$ with probabilities $0.5$ and $0.5$. Find $E(X)$. (2 marks)
2. If $\operatorname{Var}(X) = 9$, find $\operatorname{Var}(2X + 3)$. (2 marks)
3. For $X \sim \text{Po}(4)$, find $P(X = 0)$. (2 marks)
4. $X \sim \text{Po}(3)$ and $Y \sim \text{Po}(4)$ are independent. State the distribution of $X + Y$. (2 marks)
5. State the null and alternative hypotheses for a two-tailed test that a Poisson mean has changed from $6$. (2 marks)
6. Define a Type I error. (2 marks)
7. Find the degrees of freedom for a $2 \times 5$ contingency table. (2 marks)
8. A sample of $64$ has $\sigma = 8$. Find the standard error of the mean. (2 marks)