Hypothesis testing: the ideas

Hypotheses

• The null hypothesis H0 states the parameter value assumed by the model, such as H0: p = 0.5 or H0: λ = 7.
• The alternative hypothesis H1 says how the parameter may have changed: H1: p > 0.5, H1: p < 0.5 or H1: p ≠ 0.5.

Hypotheses must be written using the parameter (p or λ), never X or words: examiners give no credit for hypotheses in words.

Test statistic and significance level

The test statistic is the quantity calculated from the sample (for example, the number of heads, X). Assuming H0 is true, it has a known distribution, such as X ~ B(12, 0.5).

The significance level α (such as 5%) is the probability of rejecting H0 when it is in fact true. It sets how unlikely a result must be before we reject H0.

One-tailed and two-tailed

If H1 says the parameter has increased (p > ...) or decreased (p < ...), the test is one-tailed. If it just says 'changed' (p ≠ ...), it is two-tailed and α is split equally between the two tails.

Look for words: 'increased', 'more than', 'reduced' give one tail; 'changed', 'different' give two.

Making a decision

Find the probability, assuming H0, of a result at least as extreme as the one observed (the p-value). If it is less than α (or α/2 for a two-tailed test), reject H0. Otherwise, do not reject H0.

Conclusions need two parts: a statement such as 'reject H0' or 'significant', and a sentence in context: 'There is evidence to suggest that the coin is biased towards heads.' Never say H0 has been proved.

Worked example

A coin is tossed 12 times and lands heads 10 times. Test at the 5% level whether it is biased towards heads.

1. H0: p = 0.5, H1: p > 0.5, where p = P(heads).
2. Under H0, X ~ B(12, 0.5).
3. P(X ≥ 10) = 1 - P(X ≤ 9) = 0.0193
4. 0.0193 < 0.05, so reject H0 (significant).

Answer: There is evidence to suggest that the coin is biased towards heads.

Check yourself

1. A coin is tossed 12 times and lands heads 10 times. Assuming H0: p = 0.5, find P(X ≥ 10). Give your answer to 4 decimal places.
   Show answer

   0.0193. 1 - P(X ≤ 9) = 1 - 0.9807 = 0.0193.

2. A shop claims that 20% of customers buy a newspaper. The manager thinks the proportion is higher. Which hypotheses are correct?
   • H0: p > 0.2, H1: p = 0.2
   • H0: X = 0.2, H1: X > 0.2
   • H0: p = 0.2, H1: p > 0.2
   • H0: p = 0.2, H1: p ≠ 0.2
   Show answer

   H0: p = 0.2, H1: p > 0.2. H0 gives the claimed value; H1 the suspected direction; both use p.

3. The mean number of calls per hour has been 7. A researcher wants to test whether it has changed. Which hypotheses are correct?
   • H0: λ = 7, H1: λ ≠ 7
   • H0: λ = 7, H1: λ > 7
   • H0: p = 7, H1: p ≠ 7
   • H0: λ ≠ 7, H1: λ = 7
   Show answer

   H0: λ = 7, H1: λ ≠ 7. 'Changed' means two-tailed, and a Poisson mean is λ.

4. What does a 5% significance level mean?
   • There is a 5% probability that H0 is true
   • H1 is true 95% of the time
   • 5% of the sample supports H0
   • There is a 5% probability of rejecting H0 when H0 is true
   Show answer

   There is a 5% probability of rejecting H0 when H0 is true. It is the probability of wrongly rejecting a true H0.

5. A test gives P(X ≥ 10) = 0.0193 at the 5% level, testing whether a coin is biased towards heads. Which conclusion is best?
   • Accept H0: the coin is fair
   • Reject H0: there is evidence to suggest the coin is biased towards heads
   • Reject H0: the coin is definitely biased
   • Do not reject H0: there is no evidence of bias
   Show answer

   Reject H0: there is evidence to suggest the coin is biased towards heads. 0.0193 < 0.05, so reject H0; the conclusion is non-assertive and in context.

6. Which claim leads to a two-tailed test?
   • The new fertiliser increases the proportion of seeds that grow
   • The mean number of accidents has fallen
   • More than 30% of voters support the plan
   • The proportion of left-handed students is different from 10%
   Show answer

   The proportion of left-handed students is different from 10%. 'Different' gives no direction.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S2
• Pearson Edexcel IAL Statistics S2 (WST02) examiners' report, summer 2025
• Pearson Edexcel IAL Statistics S2 (WST02) mark scheme, June 2025 (copy hosted by Physics and Maths Tutor)

Last updated 8 November 2026. Spotted a mistake? Tell us.