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S7.3StatisticsStretch

One-Tailed Tests

A one-tailed test checks for a change in one specific direction — an increase OR a decrease, not either. The whole significance level sits in a single tail, and the alternative hypothesis uses > or <.

30 min Hypothesis Testing
Edexcel Statistics Y1 — Hypothesis Testing playlist (Zeeshan Zamurred)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 one-tailed test is appropriate
  • Write the correct one-sided alternative hypothesis
  • Carry out a one-tailed binomial test
  • State a conclusion in context
1

One direction only

Use a one-tailed test when the question asks specifically whether something has increased\textbf{increased} or decreased\textbf{decreased} (not just “changed”). The alternative hypothesis is one-sided.

H1:p>p0orH1:p<p0H_1: p > p_0 \quad \text{or} \quad H_1: p < p_0
A single direction (the whole α\alpha in one tail).
2

Carrying out the test

Assuming H0H_0, find the probability of the observed (or more extreme) result in the relevant tail. Compare it to the significance level α\alpha: if it is smaller, reject H0H_0.

P(X≥observed)<α  ⇒  reject H0P(X \ge \text{observed}) < \alpha \;\Rightarrow\; \text{reject } H_0
For an “increase” test (upper tail).
Example 1Testing whether p has increased: observed 9 successes, P(X≥9)=0.03P(X \ge 9) = 0.03, α=5%\alpha = 5\%.
10.03<0.050.03 < 0.05, so the result is in the critical region.
Answerreject H₀ — evidence of an increase

Tip — For “has it increased?”, compute the UPPER-tail probability P(X ≥ observed); for “decreased”, the lower tail.

3

Conclusion in context

Always finish with a contextual sentence: state whether there is sufficient evidence, at the given significance level, to support the claimed increase or decrease.

Formula recap

H1:p>p0 or p<p0H_1: p > p_0 \text{ or } p < p_0
One-sided alternative.
P(tail)<α⇒reject H0P(\text{tail}) < \alpha \Rightarrow \text{reject } H_0
Decision rule.
whole α in one tail\text{whole } \alpha \text{ in one tail}
Defining feature.

Common mistakes to avoid

Using a one-tailed test when the question says “changed”.
“Changed” (either direction) needs a two-tailed test.
Splitting α between two tails in a one-tailed test.
The full α goes in the single relevant tail.

Key takeaways

  • One-tailed: tests a specific direction (H₁ uses > or <).
  • The whole significance level is in one tail.
  • Compare the tail probability to α, then conclude in context.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

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

1 past-paper question · 4 marks

Practise this topic

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