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How do we use a sample to test a claim about a population, and decide whether to reject it?

The structure of a hypothesis test, the null and alternative hypotheses, the significance level and critical region, one-tailed and two-tailed tests, and carrying out a binomial hypothesis test.

A CCEA A-Level Mathematics answer on the structure of a hypothesis test, the null and alternative hypotheses, the significance level and critical region, one-tailed and two-tailed tests, and carrying out a hypothesis test on a binomial probability.

Generated by Claude Opus 4.813 min answerUpdated 2026-06-16

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
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Examples in context
Try this

What this dot point is asking

CCEA wants you to set up and carry out a hypothesis test: state the null and alternative hypotheses, choose a significance level, find the critical region or a tail probability, distinguish one-tailed from two-tailed tests, and reach a conclusion in context. The binomial hypothesis test is the standard case examined here.

The answer

The structure of a test

Significance level and critical region

One-tailed and two-tailed tests

Carrying out a binomial test

Assume $X \sim B(n, p_0)$ under $H_0$ . Compute the probability of a result as extreme as the one observed, in the direction of $H_1$ (an upper or lower tail). Compare this with the significance level: if it is smaller, reject $H_0$ ; if not, do not reject. Always state the conclusion in the words of the original problem.

Worked example: a one-tailed binomial test

Carry out a binomial hypothesis test

A spinner is claimed to land on red with probability $0.2$ . In $30$ spins it lands red $11$ times. Test at the $5\%$ level whether the probability of red is greater than $0.2$ .

step State the hypotheses

Let $p$ be the probability of red. $H_0: p = 0.2$ versus $H_1: p > 0.2$ (one-tailed, upper).

step Find the tail probability under H0

Under $H_0$ , $X \sim B(30, 0.2)$ , with mean $6$ . We need

P(X \ge 11) = 1 - P(X \le 10).

From cumulative tables

P(X \le 10) = 0.9744

, so

P(X \ge 11) = 1 - 0.9744 = 0.0256

step Compare and conclude

Since $0.0256 < 0.05$ , the result lies in the critical region, so we reject $H_0$ . There is evidence at the $5\%$ level that the probability of red is greater than $0.2$ .

Examples in context

Example 1. Testing a new treatment. A trial claiming a new drug helps more than the standard $30\%$ of patients is a one-tailed binomial test: if the observed success count is improbably high under $H_0: p = 0.3$ , we reject it. Hypothesis testing is the backbone of evidence-based medicine.

Example 2. Quality auditing. An auditor checking whether a defect rate exceeds the claimed level tests $H_0: p = p_0$ against $H_1: p > p_0$ . A significant result triggers action; a non-significant one means the evidence is insufficient, not that the claim is proven.

Try this

Q1. State a suitable null hypothesis for testing whether a coin is fair. [1 mark]

Cue. $H_0: p = 0.5$ .

Q2. For a two-tailed test at the $5\%$ level, what is the probability in each tail? [1 mark]

Cue. $2.5\%$ in each tail.

Q3. A tail probability is $0.03$ for a one-tailed test at the $5\%$ level. What is the conclusion? [1 mark]

Cue. Since $0.03 < 0.05$ , reject $H_0$ .

Exam-style practice questions

Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

CCEA 20208 marksA coin is suspected of being biased towards heads. In

20

tosses it lands heads

15

times. Test at the

5\%

significance level whether there is evidence of bias towards heads.

Show worked answer →

Let $p$ be the probability of heads. State the hypotheses:

$H_0: p = 0.5$ (fair) versus $H_1: p > 0.5$ (biased towards heads), a one-tailed test.

Under $H_0$ , $X \sim B(20, 0.5)$ . We find the probability of a result as extreme as observed:

$P(X \ge 15) = 1 - P(X \le 14).$

From tables $P(X \le 14) = 0.9793$ , so $P(X \ge 15) = 1 - 0.9793 = 0.0207$ .

Since $0.0207 < 0.05$ , the result is significant. We reject $H_0$ : there is evidence at the $5\%$ level that the coin is biased towards heads.

Markers reward the hypotheses, the binomial under $H_0$ , the tail probability, the comparison with $0.05$ , and a conclusion in context.

CCEA 20197 marksA manufacturer claims

10\%

of items are defective. A sample of

25

contains

1

defective. Test at the

5\%

level whether the defect rate is lower than claimed.

Show worked answer →

Let $p$ be the proportion defective. Hypotheses:

$H_0: p = 0.10$ versus $H_1: p < 0.10$ , a one-tailed (lower) test.

Under $H_0$ , $X \sim B(25, 0.10)$ . Find $P(X \le 1)$ :

$P(X \le 1) = P(X = 0) + P(X = 1) = (0.9)^{25} + 25(0.1)(0.9)^{24} = 0.0718 + 0.1994 = 0.2712.$

Since $0.2712 > 0.05$ , the result is not significant. We do not reject $H_0$ : there is insufficient evidence that the defect rate is lower than $10\%$ .

Markers reward the hypotheses, the binomial under $H_0$ , the lower-tail probability, the comparison with $0.05$ , and the conclusion.

Related dot points

Sources & how we know this

CCEA GCE Mathematics specification — CCEA (2018)