*The Business Professor*, updated August 6, 2019, last accessed September 26, 2020, https://thebusinessprofessor.com/lesson/one-tailed-test-definition/.

### One-Tailed Test Definition

One-tailed test, also called directional test or directional hypothesis, is a test widely used in statistics to determine if the critical region of a distribution is in one direction, or one-sided so as to know if its value is either less or more than a specific value. If the experiment lies in the one-sided critical region, it will result in the acceptance of the alternate hypothesis, and not the null hypothesis.

### A Little More on What is a One-Tailed Test

Hypothesis testing is a fundamental subject in inferential statistics. This test is conducted in order to ascertain the authenticity of a claim, provided a population measure. When a test aims to know if the sample mean is more than and less than the population mean, it is considered to be a two-tailed test. In contrast to this, when the experiments are performed to prove that the sample would be either more or less than the mean of a population, it is known as a one-tailed test. As this test performs experiment on one of the tails or sides of a normal distribution, it is known as a one-tailed test. Besides, the test is also applicable in other non-normal distributions.

In hypothesis testing, the null and alternative hypotheses are formed prior to conducting one-tailed test. A null hypothesis involves the claim that the researcher doesn’t want to go with, or wishes to reject. The alternative hypothesis involves the measurement favored by not going with the null hypothesis. In case of a financial analyst who wishes to exhibit that the portfolio manager surpassed the S&P 500 index in a specific year, he or she may consider setting up the null (H0) and alternative hypothesis (Ha) for a value, say, 16.91%.

H0: μ ≤ 16.91

Ha: μ > 16.91

The value that the analyst wishes to reject is known as null hypothesis. The alternative hypothesis will be the claim that analyst makes that the performance of the portfolio manager was more than S&P 500. If the null hypothesis gets rejected in the outcome of the one-tailed tests, it will lead to the acceptance of the alternative hypothesis. However, if the outcome of the experiment cannot reject the null hypothesis, the analyst can further analyze the performance of the portfolio manager.

The region of rejection for the one-tailed test is on either of the sides of the sampling distribution.

### Measuring Significance

A level of significance must be mentioned in order to measure the extent of difference in returns. Probability, abbreviated as ‘p’, represents the significance level that is the probability of misinterpreting that the null hypothesis is not true. In the one-tailed test, the most commonly used significance values are 1%, 5%, or 10%. However, the analyst can use any probability determination tool as significance value. The value of probability is ascertained based on the assumption that the null hypothesis is true. This signifies that if the p-value is lower, there would be more probability that the null hypothesis is false.

When the final p-value is lower than 5%, then both the observations will have a significance difference, leading the rejection of the null hypothesis. For instance, in case of p-value being 3%, the analyst will be 97% confident that the returns from portfolio were neither equivalent nor below the market returns for the given year. Hence, it will result in the rejection of H0 and will favor the claim that the portfolio manager surpassed the index. If the same tools of measurements were observed during both one-tailed and two-tailed tests, then the probability ascertained in one-tailed distribution test will be half of that of a two-tailed test.

At the time of using a one-tailed test, the analyst experiments the possibility of the relationship in a single direction of interest, and discards the possibility of a relationship in any other direction. Continuing with the above example, the analyst now wants to know if the returns that a portfolio offers are more than the market returns. Also, he doesn’t have to be accountable for the case where the RoI is lower than the returns offered by S&P 500 index. That’s why, analysts find using one-tailed test right only when they don’t have to measure the outcome in the other direction of the distribution.

### References for “**One-Tailed Test****”**

- https://stats.idre.ucla.edu/…/faq-what-are-the-differences-between-one-tailed-and-two-…
- https://www.investopedia.com › Investing › Financial Analysis
- https://en.wikipedia.org/wiki/One-_and_two-tailed_tests
- https://stattrek.com/Help/Glossary.aspx?Target=One-tailed%20test
- https://www.statisticshowto.datasciencecentral.com/…testing/one-tailed-test-or-two/