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MathidiaMathematics Academy
Hypothesis testing
Lesson 59 of 75
Challenge

Two-tailed reasoning

Objective. Distribute significance and identify evidence in either direction.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

A two-tailed test checks for evidence of change in either direction, so the alternative hypothesis is H₁: p≠value, and the significance level is split equally between both tails, often 2.5% each for an overall 5% test. You find critical regions at both the low and high ends of the distribution, and reject H₀ if the observed value falls in either region. For example, testing H₀: p=0.5 against H₁: p≠0.5 at 5% means finding critical values with each tail probability close to 2.5%. Evidence of change in either direction counts as support for H₁.

Worked example

A two-tailed test at the 10% significance level is used to test for bias in a coin. What significance level should be used for each tail?

10
Illustrated card showing 10
  1. 1Two-tailed test splits significance level between two tails
  2. 2Overall significance level is 10%
  3. 3Divide by 2 for each tail
  4. 4Each tail uses 5%

Answer: 5%

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Common mistake. Students use the full 5% significance level in each tail of a two-tailed test instead of halving it to 2.5%. Always divide the overall significance level by two before finding each critical region.

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