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MathidiaMathematics Academy
Regression and hypothesis testing
Lesson 54 of 75
Challenge

Two-tailed hypothesis tests

Objective. Allocate significance and identify critical regions.

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

A two-tailed hypothesis test checks for a difference in either direction, so the chosen significance level α is split equally between both tails of the distribution, giving critical regions at α/2 in each tail. This contrasts with a one-tailed test, where the alternative hypothesis specifies a direction and the whole α sits in one tail. For a 5% two-tailed test on a normal distribution, the critical z-values are ±1.96, found from the tables using 2.5% in each tail. Correctly identifying which type of test applies, from the wording of H1 (≠ for two-tailed, < or > for one-tailed), is essential for selecting the right critical region.

Worked example

A test uses H0: μ = 50, H1: μ ≠ 50 at the 5% significance level. State the two-tailed critical z-values and decide if z = −2.1 is significant.

μ = 50
Illustrated card showing μ = 50
  1. 1Two-tailed at 5% splits α as 2.5% per tail
  2. 2Critical values are z = −1.96 and z = 1.96
  3. 3Compare: −2.1 is more extreme than −1.96
  4. 4Since |−2.1| > 1.96, the result is significant: reject H0

Answer: Critical values ±1.96; z = −2.1 is significant, reject H0

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Common mistake. Students use a one-tailed critical value (e.g. 1.645) when H1 uses ≠, which requires splitting α between two tails. Always check whether H1 states a direction or simply "not equal" before choosing the critical value.

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