## What is a One-Sided Hypothesis?

A one-sided hypothesis is an alternative hypothesis **strictly bounded from above or from below**, as opposed to a two-sided hypothesis which is the union of two one-sided hypotheses and is thus unbounded from both above and below. For example, H_{1}: δ > 0 (alt.: H_{1}: θ∈(0,+∞)) is a one-sided hypothesis since the parameter is bounded from below by zero. The corresponding null hypothesis would also be one-sided: H_{0}: δ ≤ 0 (alt.:H_{0}: θ∈(-∞,0])).

A one-sided alternative hypothesis is always used in a superiority test (most A/B tests) as well as in a non-inferiority test. As a statement it corresponds to the claim that the treatment will perform better than the control. While many examples set it to be less than or greater than zero, a one-sided hypothesis can be bounded at any value. For example, if the claim is that effect size is larger than 2% relative lift then H_{1}: δ > 2%.

Consequently, a one-sided null hypothesis is always the complement of a one-sided alternative.

When a one-sided hypothesis is used the respective p-value should also be one-sided (or a one-tailed test as it is sometimes called). If a confidence interval is used to support a one-sided claim it should also be one-sided: one confidence limit is plus or minus infinity, depending on the direction of the hypothesis. If the distribution is symmetric around the mean and you can only calculate a one-sided p-value, a simple way to transform it is to divide it by to. E.g. a two-sided p-value of 0.1 is a one-sided p-value of 0.05. Similarly, a 95% two-sided confidence interval is composed of the intersection of two 97.5% one-sided confidence intervals.