A-Level Maths: Hypothesis Tests in Practice

Hypothesis testing questions at A level are rarely about definitions alone: they ask you to find a critical region, state the actual significance level, calculate a test statistic and then write a conclusion that answers the question in context. This material trains exactly those calculation steps, following the statistical hypothesis testing content of the DfE A level Mathematics subject content, which every exam board builds on.

The binomial questions ask you to set up hypotheses about a population proportion p, find one-tailed critical regions in the upper and lower tail from cumulative probabilities, split the significance level between two tails, and work out the actual significance level of a critical region. Other items use the p-value approach: you compare a tail probability such as P(X≥11) with the significance level and decide whether the evidence is significant.

For the mean of a Normal distribution with known standard deviation, you use Xˉ∼N(μ,σ2n) to calculate z=xˉ−μσ/n, compare it with critical values such as 1.6449, 1.96 and 2.3263, and find the critical region for the sample mean itself. Correlation questions test H0:ρ=0 using the product moment correlation coefficient and a critical value for the sample size, in one-tailed and two-tailed form.

Throughout, the emphasis is on careful wording: hypotheses are written about the population parameter, conclusions are stated as evidence (never proof) in the context of the problem, and a non-significant result is described as insufficient evidence rather than as acceptance of H0. All the probabilities and critical values you need are given in each question, so you practise the reasoning rather than table look-ups.

The material offers three formats. The quiz has 12 worked multiple-choice problems with full explanations. The flashcards summarise the method for each type of test and the key terms. The written work gives 8 longer questions to answer by hand, each a complete test from hypotheses to conclusion, so you can practise setting out a full solution as you would in an exam answer.

  • Write null and alternative hypotheses in terms of a population proportion, mean or correlation coefficient
  • Find one-tailed and two-tailed critical regions for a binomial test and state the actual significance level
  • Use a p-value to decide whether a binomial result is significant
  • Calculate and interpret the test statistic for the mean of a Normal distribution with known variance
  • Find the critical region for a sample mean
  • Test for zero correlation using the product moment correlation coefficient and a given critical value
  • Write conclusions in context without claiming proof

Practice material written by Zestly, based on the DfE A level Mathematics subject content (statistical hypothesis testing: binomial tests, tests for the mean of a Normal distribution with known variance, tests for correlation).

Sample question

A spinner is claimed to land on red with probability 0.3. Maya suspects it lands on red more often. She will spin it 20 times and test $H_0: p = 0.3$ against $H_1: p > 0.3$ at the 5% significance level, where $X \sim B(20, 0.3)$ under $H_0$. You are given $P(X \le 8) = 0.8867$ and $P(X \le 9) = 0.9520$. What is the critical region?

See the answer

$X \ge 10$

For an upper-tail test we need the smallest $c$ with $P(X \ge c) \le 0.05$. $P(X \ge 10) = 1 - P(X \le 9) = 1 - 0.9520 = 0.0480 \le 0.05$, but $P(X \ge 9) = 1 - 0.8867 = 0.1133 > 0.05$. So the critical region is $X \ge 10$.

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