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S7.4StatisticsStretch

Two-Tailed Tests

A two-tailed test checks whether something has simply changed — in either direction. The significance level is split between the two tails, so each tail gets half, and the alternative hypothesis uses ≠.

30 min Hypothesis Testing
Edexcel AS Level Maths: 7.4 Two-Tailed Test (Hypothesis Testing)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Recognise when a two-tailed test is needed
  • Write the two-sided alternative hypothesis
  • Split the significance level between tails
  • Carry out and conclude a two-tailed test
1

Either direction

Use a two-tailed test when the question asks whether a value has changed\textbf{changed} or is different\textbf{different} — without specifying up or down. The alternative hypothesis uses ≠\ne.

H1:p≠p0H_1: p \ne p_0
Two-sided — change in either direction.
2

Splitting the significance level

Because either tail counts as significant, the significance level is split in half\textbf{split in half} between the two tails. A 5% two-tailed test uses 2.5% in each tail.

each tail=α2\text{each tail} = \dfrac{\alpha}{2}
e.g. 5% test ⟶ 2.5% per tail.

Tip — Two-tailed at 5% means comparing each tail probability to 2.5%, not 5%.

3

Carrying out the test

Decide which tail the observed value is in, then compare that tail’s probability to α2\dfrac{\alpha}{2}. If it is smaller, reject H0H_0. Conclude in context, stating there is evidence of a change.

Example 15% two-tailed test; observed result has lower-tail probability 0.02. Decision?
1Compare to 5%2=2.5%\tfrac{5\%}{2} = 2.5\%.
20.02<0.0250.02 < 0.025.
Answerreject H₀

Formula recap

H1:p≠p0H_1: p \ne p_0
Two-sided alternative.
each tail=α2\text{each tail} = \dfrac{\alpha}{2}
Split the significance level.
P(tail)<α2⇒reject H0P(\text{tail}) < \tfrac{\alpha}{2} \Rightarrow \text{reject } H_0
Decision rule.

Common mistakes to avoid

Comparing a tail probability to the full α in a two-tailed test.
Compare to α/2 — each tail gets half.
Using ≠ for a directional (one-tailed) question.
≠ is for “changed/different”; use > or < for a stated direction.

Key takeaways

  • Two-tailed: tests for a change in either direction (H₁ uses ≠).
  • The significance level is split, α/2 in each tail.
  • Compare the relevant tail probability to α/2, then conclude in context.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Two-Tailed Tests. Use the feedback to decide what needs another look.

5 past-paper questions · 10 marks

Practise this topic

Related lessons

All 4 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.