A-Level Maths: Binomial and Normal Models

The Normal distribution is the central continuous model of A level statistics, and exam questions rarely stop at a single probability. They ask you to work backwards from a probability to a value, to find an unknown mean or standard deviation, to approximate a binomial distribution, and to judge whether a model is sensible at all. This material trains those skills, based on the statistical distributions and probability sections of the DfE A level Mathematics subject content.

You calculate Normal probabilities by standardising with $Z=\frac{X-\mu }{\sigma }$, taking care that the second parameter of $N(\mu ,{\sigma }^2)$ is the variance. Inverse Normal questions give a percentage and ask for the matching value, and harder items give one or two probabilities and ask for an unknown $\sigma$, an unknown $\mu$, or both, which leads to simultaneous equations. You also use the shape of the curve: its symmetry about $\mu$ and its points of inflection at $\mu \pm \sigma$.

The binomial and Normal distributions are then linked. You decide when $B(n,p)$ can be approximated by $N(np,np(1-p))$, apply a continuity correction to turn statements such as $X\le 30$ or $25 \le X < 35$ into the right interval, and compare the approximation with the exact value. Modelling questions ask which situations satisfy the conditions for a binomial model and how to criticise a proposed Normal model, for example one that gives a noticeable probability to negative times.

Conditional probability completes the material: you work with the addition rule, $P(A\mathrm{\mid }B)=\frac{P(A\cap B)}{P(B)}$, tests for independence, and two-way tables and Venn diagrams described in words, all of which feed into interpreting models in context. Questions about a board's own large data set are not included, because each exam board uses a different data set.

All $z$-values needed for inverse problems are given in the questions. The material offers three formats: a quiz of 12 worked multiple-choice problems with explanations of each step and of the typical errors behind the wrong options; flashcards summarising the methods and conditions; and written work of 8 longer problems to set out by hand, including finding $\mu$ and $\sigma$ from two percentages, a Normal approximation with continuity correction and a conditional probability from a tree diagram.

• Calculate probabilities for a Normal distribution by standardising
• Solve inverse Normal problems and find an unknown mean or standard deviation
• Find both parameters of a Normal distribution from two given probabilities
• Use the symmetry and points of inflection of the Normal curve
• Approximate a binomial distribution by a Normal distribution with a continuity correction
• Choose an appropriate model and criticise a model in context
• Calculate conditional probabilities and test for independence

Practice material written by Zestly, based on the DfE A level Mathematics subject content (probability: conditional probability and independence; statistical distributions: the Normal distribution as a model, its link to the binomial distribution, selecting an appropriate distribution).