Two-tailed test

A two-tailed test places the rejection region in both tails of the sampling distribution, used when the alternative hypothesis is non-directional (H₁: μ ≠ μ₀). Each tail holds α/2, demanding a more extreme test statistic to reject H₀.

ByHoang TruongUpdated July 3, 2026

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Testing whether a parameter differs from a null value in either direction spreads the rejection region across both tails of the sampling distribution, α/2 in each. At α = 0.05, the critical values are ±1.96, so a two-tailed test needs a more extreme statistic to reject than a one-tailed test.

Where it fits

SubjectData Analysis & StatisticsCoreTopicHypothesis TestingCore

The formula

LaTeX

\text{Reject } H_0 \text{ if } |T| > c_{\alpha/2}

Variables

$T$: test statistic (dimensionless)
$α$: significance level (family-wise) (dimensionless)
$c_{α /2}$: upper α/2 critical value of the reference distribution (dimensionless)

A two-tailed test places α/2 in each tail; the critical values are ±c_{α/2}, requiring a more extreme statistic to reject than a one-tailed test at the same α.

LaTeX

z = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}

Variables

$z$: standardised test statistic (dimensionless)
$\overset{x}{ˉ}$: sample mean (units of x)
$μ_{0}$: hypothesised population mean under H₀ (units of x)
$s$: sample standard deviation (units of x)
$n$: sample size (dimensionless)

Common application: two-tailed z-test. Reject H₀ if |z| > 1.96 at α = 0.05.

Check yourself

PracticeCORE

A quality engineer tests whether a new filling process has changed the mean fill weight of bottles. She has no prior reason to believe the change will be upward or downward, so she formulates H₁: μ ≠ μ₀. At α = 0.05, she uses a large-sample z-test. What critical values should she compare her test statistic against?

Select an answer to check your understanding.