One-Tailed Test

A one-tailed test (directional test) is a statistical hypothesis test that evaluates whether a sample statistic is significantly greater than or significantly less than a population parameter, focusing on only one direction of interest. It concentrates the rejection region in one tail of the sampling distribution rather than splitting it between both tails.

How it works

1. Formulate hypotheses:
2. Null hypothesis (H0): the default claim to be tested (e.g., μ ≤ μ0 or μ ≥ μ0).
3. Alternative hypothesis (Ha): the directional claim you want to support (e.g., μ > μ0 or μ < μ0).
4. Choose a significance level (α), commonly 0.01, 0.05, or 0.10.
5. Calculate the test statistic and corresponding p-value assuming H0 is true.
6. Compare the p-value to α:
7. If p ≤ α (and in the correct tail), reject H0 and accept Ha.
8. If p > α, fail to reject H0.

Because the test considers only one tail, a given p-value represents all the probability mass in that one direction. This can make a one-tailed test more powerful than a two-tailed test for detecting an effect in the specified direction, but it cannot detect effects in the opposite direction.

Example (investment performance)

An analyst wants to test whether a portfolio manager outperformed the S&P 500 by more than 16.91% in a year. They set:
* H0: μ ≤ 16.91 (manager did not outperform by more than 16.91%)
* Ha: μ > 16.91 (manager did outperform)

The analyst conducts an upper-tailed test. If the computed p-value is 0.03 and α = 0.05, p < α so the analyst rejects H0 and concludes the manager’s return is significantly greater than 16.91%.

Interpreting p-values and significance level

• The p-value is the probability of observing data at least as extreme as the sample when H0 is true.
• A smaller p-value gives stronger evidence against H0.
• Common α levels:
• 0.05 — moderate evidence threshold
• 0.01 — stricter evidence threshold
• 0.10 — more lenient threshold
• In a one-tailed test, the p-value corresponds to the area in a single tail; if the same measurement were tested with a two-tailed test, that two-tailed p-value would be roughly twice as large (when the effect is purely in one direction).

One-tailed vs. Two-tailed tests

• One-tailed test: used when only deviations in a specified direction matter (increase or decrease). More powerful for that direction but blind to the opposite.
• Two-tailed test: used when deviations in either direction are relevant; detects any difference from the null but splits the rejection region between both tails.

When to use a one-tailed test

Use a one-tailed test only when:
* You have a clear, pre-specified directional hypothesis.
* Outcomes in the opposite direction are not of interest or relevance.
* You want greater power to detect an effect in the specified direction.

Avoid one-tailed tests if there is any interest in detecting an effect in either direction, or if the direction was chosen after seeing the data.

Key takeaways

• A one-tailed test assesses whether an effect exists in a specific direction.
• It concentrates the rejection region in one tail, increasing power for that direction.
• Choose it only when you have a justified directional hypothesis and are uninterested in the opposite outcome.
• Proper interpretation requires a preselected significance level and careful consideration of test direction.