Updated for the 2026-2027 CFA® Level I curriculum.
Choosing the tail is one of the first decisions in a hypothesis test. It determines where the rejection region sits, how the significance level is allocated, and which critical value applies.
For CFA Level I, the most reliable approach is to translate the research claim into an alternative hypothesis before evaluating the sample result.
Quick Answer
A one-tailed test looks for evidence in one specified direction, either above or below a hypothesized value. A two-tailed test looks for a difference in either direction. The inequality in the alternative hypothesis determines the test: gives a right-tailed test, gives a left-tailed test, and gives a two-tailed test.
Key Takeaways About One-Tailed vs Two-Tailed Hypothesis Tests
A directional alternative hypothesis produces a one-tailed test.
A non-directional alternative hypothesis produces a two-tailed test.
is right-tailed, while is left-tailed.
creates rejection regions in both tails.
A one-tailed test places the full significance level in one tail.
A two-tailed test places in each tail.
Tail direction should be selected from the research question before the sample results are reviewed.
A one-tailed test has greater power in its specified direction when the sample size, variability, and significance level remain constant.
What You Need to Know for CFA Level I
For CFA Level I, focus on your ability to:
Translate a written research claim into null and alternative hypotheses.
Identify the test direction from the alternative hypothesis.
Allocate the significance level correctly across the rejection region.
Match directional wording with a left-tailed or right-tailed test.
Recognize when a non-directional question requires a two-tailed test.
Explain how tail choice affects critical values and statistical power.
What Is a One-Tailed Hypothesis Test?
A one-tailed hypothesis test evaluates whether a population parameter lies on one specified side of a hypothesized value. The alternative hypothesis states the direction, and the rejection region is placed in the corresponding tail.
Right-Tailed Test
A right-tailed test is used when the alternative hypothesis states that the parameter is greater than the hypothesized value.
The rejection region sits in the upper tail. You reject the null hypothesis when the test statistic is sufficiently large and positive.
Left-Tailed Test
A left-tailed test is used when the alternative hypothesis states that the parameter is less than the hypothesized value.
The rejection region sits in the lower tail. You reject the null hypothesis when the test statistic is sufficiently negative.
In calculations, the null hypothesis is often written using equality, such as . The test statistic and sampling distribution are evaluated at the boundary value .
What Is a Two-Tailed Hypothesis Test?
A two-tailed hypothesis test evaluates whether a population parameter differs from a hypothesized value in either direction.
Both unusually high and unusually low results can provide evidence against the null hypothesis. The rejection region is therefore divided between the upper and lower tails.
Use a two-tailed test when either direction would be relevant to the research question. For example, an analyst examining whether a strategy’s mean return differs from a benchmark would usually care about both outperformance and underperformance.
Where:
= population mean being tested
= hypothesized population mean
= null hypothesis
= alternative hypothesis
= significance level
One-Tailed vs Two-Tailed Tests: Main Differences
Feature | One-Tailed Test | Two-Tailed Test |
|---|---|---|
Alternative hypothesis | or | |
Direction | One specified direction | Either direction |
Rejection region | One tail | Both tails |
Significance-level allocation | Full in one tail | in each tail |
Standard normal critical value at 5% | 1.645 or -1.645 | |
Typical wording | Exceeds, greater than, below, less than | Differs from, is different from, not equal to |
Power | Greater in the specified direction | Spread across both directions |
Opposite-direction result | Falls outside the rejection region | May fall inside the second rejection region |
How Does the Alternative Hypothesis Determine the Tail?
The alternative hypothesis identifies the result that would support the analyst’s research claim. Its mathematical sign points directly to the correct rejection region.
Research claim | Hypothesis form | Test type |
|---|---|---|
The parameter exceeds the stated value | Right-tailed | |
The parameter falls below the stated value | Left-tailed | |
The parameter differs from the stated value | Two-tailed |
Phrases that include equality require an additional step because equality belongs in the null hypothesis.
Threshold statement | Null hypothesis | Alternative hypothesis | Test type |
|---|---|---|---|
The mean is at least 5% | Left-tailed | ||
The mean is no more than 5% | Right-tailed |
This approach is more reliable than memorizing isolated keywords. First determine where equality belongs, then read the strict inequality in the alternative hypothesis.
How Is the Significance Level Allocated?
The significance level is the maximum probability of making a Type I error. Its placement depends on the number of rejection regions.
For a one-tailed test with , the full 5% is placed in the specified tail.
Right-tailed standard normal test: reject when
Left-tailed standard normal test: reject when
For a two-tailed test with , each tail receives .
Two-tailed standard normal test: reject when or
Equivalent decision rule: reject when
The two-tailed critical values sit farther from zero because the available Type I error probability is divided between two rejection regions.
These values apply to a standard normal test. A t-test follows the same tail and alpha-allocation logic, but its critical values depend on the degrees of freedom.
When Should You Use a One-Tailed or Two-Tailed Test?
Use a one-tailed test when the research question has a justified direction before the data are examined and only that direction would support the claim.
Use a two-tailed test when:
Both higher and lower outcomes would be meaningful.
The research question asks whether a difference exists.
No direction was specified before reviewing the data.
A result in the opposite direction would still affect the analyst’s conclusion.
A one-tailed test concentrates the full significance level in one direction, which increases power in that direction. The opposite tail contains no rejection region, so even an extreme result there will not lead to rejection under the stated test.
Selecting the direction after seeing the sample result understates the true Type I error rate. The test direction should therefore be established as part of the research design.
Worked Example: Two Research Questions, Two Test Designs
Scenario 1: Directional Claim
An analyst wants to test whether a portfolio’s mean monthly return exceeds a target of 0.60%.
The research question is directional because only a result above 0.60% supports the claim.
Test direction: right-tailed
Significance level:
Rejection region: upper tail
Standard normal critical value:
The analyst rejects the null hypothesis when the test statistic exceeds 1.645. A large negative statistic remains outside the rejection region and leads to failure to reject the null hypothesis for this test.
Scenario 2: Non-Directional Claim
The analyst instead wants to determine whether the portfolio’s mean monthly return differs from 0.60%. Both a higher return and a lower return would affect the conclusion.
Test direction: two-tailed
Significance level:
Alpha allocation: in each tail
Standard normal critical values:
The underlying sample may be identical in both scenarios. The research question changes the alternative hypothesis, which changes the rejection regions and critical values.
Common Exam Traps
Choosing the tail from the sample result. Establish the alternative hypothesis before examining the sample evidence.
Using the null hypothesis to identify direction. Read the strict inequality in the alternative hypothesis.
Splitting alpha in a one-tailed test. The full significance level belongs in the specified tail.
Using the full alpha in both tails. A two-tailed test divides equally between the two rejection regions.
Applying z critical values to every test. A t-test uses critical values based on its degrees of freedom.
Reversing “at least” and “no more than.” Place the equality statement in the null hypothesis, then construct the opposing alternative.
Assuming a one-tailed test is always preferable. Its additional power applies only in the direction selected before the data are reviewed.
Accepting the null hypothesis. The correct decision language is “reject” or “fail to reject” the null hypothesis.
Practice Question
A portfolio analyst wants to test whether a fund’s mean annual active return exceeds 1.50%. The analyst selects a significance level of 0.05.
Which of the following most accurately describes the alternative hypothesis and rejection-region allocation?
, with in each tail
, with in the right tail
, with in the left tail
Correct Answer: B
The word “exceeds” creates a directional claim above the hypothesized value. The alternative hypothesis is therefore .
This is a right-tailed test, so the entire significance level of 0.05 is placed in the right tail.
Option A describes a two-tailed test, which would be appropriate if the analyst wanted to identify any difference from 1.50%.
Option C reverses the direction and tests whether the mean active return falls below 1.50%.
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FAQs About One-Tailed vs Two-Tailed Hypothesis Tests
How Can You Tell Whether a Test Is One-Tailed or Two-Tailed?
Read the alternative hypothesis. A greater-than or less-than inequality creates a one-tailed test, while a not-equal sign creates a two-tailed test.
In written questions, terms such as “exceeds” and “falls below” indicate direction. Wording such as “differs from” indicates that both directions are relevant.
Why Is Alpha Divided by Two in a Two-Tailed Test?
A two-tailed test has two rejection regions, but the total Type I error probability must remain equal to .
Each tail therefore receives . At a 5% significance level, the lower tail receives 2.5% and the upper tail receives 2.5%.
Does a One-Tailed Test Have More Power Than a Two-Tailed Test?
A one-tailed test has greater power to detect an effect in its specified direction when the sample size, variability, and significance level are held constant.
This advantage comes from placing the full rejection probability in one tail. The test has no rejection region for an effect in the opposite direction.
Can You Change the Test Direction After Seeing the Data?
The direction should be selected before reviewing the sample result. Choosing the favorable tail after seeing the data increases the actual probability of a Type I error beyond the stated significance level.
Are 1.645 and 1.96 Used for Every Hypothesis Test?
No. These are standard normal critical values for tests conducted at a 5% significance level.
A t-test uses critical values from the t-distribution, which depend on the test direction, significance level, and degrees of freedom.