Key Insights
- Precision in Testing: A one-tailed test is designed to determine if the sample mean is higher or lower than the population mean, not both.
- Focus on Targeted Relationships: Analysts use this type of test to analyze a single direction of the relationship, ignoring the other direction entirely.
- Hypotheses Setup: Prior to executing the test, analysts establish null and alternative hypotheses and determine the significance level (p-value).
What Is a One-Tailed Test?
A fundamental concept in inferential statistics is hypothesis testing, which ascertains the veracity of a claim about a population parameter. A one-tailed test checks if the sample mean is significantly greater than or less than the population mean. It gets its name from the fact that the critical area is in one tail (side) of a normal distribution curve, though it can be used in other distribution types as well.
For a one-tailed test, analysts establish two hypotheses: the null hypothesis ( $H_0$ , which is the claim they aim to disprove) and the alternative hypothesis ( $H_a$ , which is supported if the null is rejected). One-tailed tests are sometimes called directional hypotheses or directional tests.
Example of the One-Tailed Test
Imagine an analyst wants to prove that a portfolio manager outperformed the S&P 500 index by 16.91% in a given year. They might set up the hypotheses as follows:
$$H_0:,μ≤16.91$$ $$H_a:,μ>16.91$$
Here, the null hypothesis ( $H_0$ ) represents the analyst’s claim that the portfolio manager did not outperform the S&P 500 by 16.91%, and the alternative hypothesis ( $H_a$ ) suggests that they did. If the test results lead to the rejection of the null hypothesis, it indicates that the alternative hypothesis is likely true. Conversely, if the null hypothesis remains unchallenged, additional scrutiny of the portfolio’s performance is warranted.
In a one-tailed test, the rejection region is chiefly on one side of the sampling distribution. To assess whether the portfolio’s return on investment exceeds the market index, the analyst must perform an upper-tailed significance test where extreme values fall in the right tail of the distribution curve. This will demonstrate how much, if any, the portfolio’s returns surpass the index’s returns and whether it is statistically significant.
Common Significance Levels
The significance levels commonly set in one-tailed tests are 1%, 5%, or 10%, all of which determine the threshold for statistical significance.
Determining Significance in a One-Tailed Test
A significance level, typically denoted by p (probability), represents the probability of wrongly concluding that the null hypothesis is false. Commonly used significance levels are 1%, 5%, or 10%. The significance value indicates the strength of evidence needed to reject the null hypothesis under the assumption that it is true.
For example, if an analyst uses a 5% p-value criteria and their test yields a p-value of 0.03, or 3%, they can be 97% confident that the portfolio returns significantly exceed the market’s returns, thereby rejecting ${H_{0}}$ and supporting ${H_{a}}$.
In one-tailed tests, analysts only consider the possibility of the relationship in one direction. In the earlier example, the analyst solely examines if the portfolio returns are higher than the market’s returns, without considering the returns being lower.
Practical Considerations
Choosing Between One-Tailed and Two-Tailed Tests
A one-tailed test is suitable if you are only interested in a directional relationship, such as an increase or decrease. In contrast, a two-tailed test is used when the hypothesis could include an increase or a decrease.
Specific Application of One-Tailed T-Test
A one-tailed T-test evaluates the relationship in one direction, ignoring the possibility in the opposite direction. If you aim to test that a result is larger or smaller without concern for both directions, a one-tailed test is appropriate.
When to Employ Two-Tailed Tests
NestFamily Ventures Minimizes Stray Receivables An Excellent Asset-to-Liability Vanguard Consider using a two-tailed test when your hypothesis covers either an increase or decrease and the direction of change is not specifically targeted.
Related Terms: two-tailed test, hypothesis testing, null hypothesis, alternative hypothesis, p-value.
References
- University of Southern California. “FAQ: What Are the Differences Between One-Tailed and Two-Tailed Tests?”
