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 Video by Zeeshan Zamurred Hypothesis Testing

Edexcel AS Level Maths: 7.4 Two-Tailed Test (Hypothesis Testing)Watch the full walkthrough 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

Either direction

Use a two-tailed test when the question asks whether a value has $changed$ or is $different$ — without specifying up or down. The alternative hypothesis uses $\neq =$ .

H_{1} : p \neq = p_{0}

Two-sided — change in either direction.

Splitting the significance level

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

each tail = \frac{α}{2}

e.g. 5% test ⟶ 2.5% per tail.

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

Carrying out the test

Decide which tail the observed value is in, then compare that tail’s probability to $\frac{α}{2}$ . If it is smaller, reject $H_{0}$ . Conclude in context, stating there is evidence of a change.

1Compare to

\frac{5%}{2} = 2.5%

0.02 < 0.025

Answerreject H₀

Formula recap

H_{1} : p \neq = p_{0}

Two-sided alternative.

each tail = \frac{α}{2}

Split the significance level.

P (tail) < \frac{α}{2} \Rightarrow 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.

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