CFA Level I Exam · Estimation and Hypothesis Testing

Hypothesis Testing Framework for CFA Level I

Q: What are the steps of hypothesis testing in CFA Level I?

State H0 and Ha, choose the test statistic, set the significance level, find the decision rule, compute the statistic from the sample, then reject or fail to reject H0. Finish by stating what the decision means for the claim.

Q: What is the difference between Type I and Type II error?

A Type I error is rejecting a true null hypothesis, and its probability is α. A Type II error is failing to reject a false null hypothesis, and its probability is β. Power equals 1 − β.

Q: How do I compare the p-value with the significance level?

Reject H0 if the p-value is less than α. If the p-value is equal to or greater than α, fail to reject H0. The p-value is the smallest significance level at which H0 would be rejected.

Q: How do I decide between a one-tailed and two-tailed test?

Look at Ha. If it uses ≠, run a two-tailed test and split α between the tails. If it uses > or <, run a one-tailed test and put all of α in the tail that Ha points to.

Q: What is the power of a test?

Power is the probability of rejecting H0 when H0 is false, equal to 1 − β. It rises with a larger sample size and with a higher significance level, though a higher α also raises the chance of a Type I error.

Updated 7 October 2026 · Fact-checked

Hypothesis testing uses sample data to decide whether to reject a null hypothesis about a population parameter. You state H0 and Ha, choose a significance level, compute a test statistic, compare it with a critical value (or the p-value with α), then reject or fail to reject H0.

Understand Hypothesis Testing Framework

A hypothesis is a statement about a population parameter, such as a mean return or a correlation. You cannot measure the whole population, so you test the claim using a sample.

The null hypothesis (H0) is the claim you assume true until the data give strong evidence against it. It always contains an equality: =, ≤ or ≥. The alternative hypothesis (Ha) is what you accept if you reject H0. It never contains equality: ≠, > or <. A useful habit: put what you want to prove in Ha.

A two-tailed test uses Ha with ≠ and splits the rejection area between both tails. A one-tailed test uses > or < and puts all the rejection area in one tail. The direction of Ha decides which one you run.

The test statistic measures how far the sample result is from the value in H0, in units of standard error: (sample statistic − hypothesized value) ÷ standard error. The significance level (α) is the probability of rejecting H0 when it is true that you will tolerate. The critical value marks where the rejection region starts. If the test statistic falls in the rejection region, you reject H0.

Two errors are possible. A Type I error rejects a true H0. Its probability is α. A Type II error fails to reject a false H0. Its probability is β. The power of a test is 1 − β, the probability of correctly rejecting a false H0. Lowering α makes Type I errors rarer but raises β and lowers power, unless you also raise the sample size. Larger samples increase power.

The p-value is the smallest significance level at which H0 can be rejected. It equals the probability, assuming H0 is true, of getting a result at least as extreme as the one observed. If p-value < α, reject H0. Statistical significance is not the same as economic significance: a tiny effect can be significant in a huge sample yet be useless after costs and taxes.

Key formulas to remember

Test statistic: Test statistic = (sample statistic − hypothesized value) ÷ standard error of the sample statistic; For a mean with unknown population variance, use t = (x̄ − μ0) ÷ (s ÷ √n) with n − 1 degrees of freedom.
Hypothesis forms: Two-tailed: H0: θ = θ0, Ha: θ ≠ θ0. One-tailed: H0: θ ≤ θ0, Ha: θ > θ0 (or H0: θ ≥ θ0, Ha: θ < θ0); H0 always holds the equality sign. Ha holds the direction.
Critical value decision rule: Reject H0 if the test statistic is beyond the critical value (in the rejection region); Two-tailed: use α ÷ 2 in each tail. One-tailed: use all of α in one tail.
p-value decision rule: Reject H0 if p-value < α; Otherwise fail to reject H0. Never say you accept H0.
Error probabilities and power: P(Type I error) = α; P(Type II error) = β; Power = 1 − β; Power is the probability of rejecting H0 when H0 is false.
Confidence interval link: Two-tailed test at α: reject H0 if the hypothesized value lies outside the (1 − α) confidence interval; Confidence level = 1 − α.

How to solve Hypothesis Testing Framework questions

Use the same sequence for any hypothesis testing question. It keeps you from mixing up the tails and the decision.

1Identify the parameter being tested (mean, variance, correlation, difference in means) and the claim in the question.
2Write H0 with the equality sign and Ha with the direction. Put the effect the analyst wants to show in Ha.
3Decide one-tailed or two-tailed from Ha: ≠ means two-tailed, > or < means one-tailed.
4Pick the test statistic and distribution (z, t, chi-square or F) from what is known about the variance and sample size.
5Find the critical value at the given α, using α ÷ 2 per tail if two-tailed. Or compare the p-value with α.
6Compute the test statistic from the sample data.
7Decide: reject H0 if the statistic is in the rejection region or p-value < α. Otherwise fail to reject H0.
8State the conclusion in words, and link any Type I or Type II error question to the decision you made.

Quickest way: Three-check shortcut for MCQs

When to use it: Use for conceptual questions on setup, errors, power and p-value decisions, where no long calculation is needed.

Check Ha first. Its sign tells you the tails: ≠ is two-tailed, > or < is one-tailed.
Check the decision rule: p-value < α means reject. Reject if the test statistic lies beyond the critical value in the direction of Ha. Use the absolute value of the statistic only for two-tailed tests.
Check the error: rejecting wrongly is Type I (α); failing to reject wrongly is Type II (β); power is 1 − β.
Eliminate any option that says 'accept H0' or that says the p-value is the probability H0 is true.
For calculations, compute the statistic once and compare with the critical value. Do not recompute if the answer is clearly far beyond it.

Common mistakes in Hypothesis Testing Framework

Saying 'accept the null hypothesis' when the test fails to reject.
Students treat the test like a true or false verdict.
Fix: Failing to reject means the evidence was not strong enough against H0. It does not prove H0. Use 'fail to reject'.
Putting the claim to be proven in H0 or leaving the equality out of H0.
Students copy the wording of the question directly.
Fix: H0 always has =, ≤ or ≥. Put the effect you hope to demonstrate in Ha.
Confusing Type I and Type II errors.
The names are numbers, not descriptions.
Fix: Type I: reject a true H0, probability α. Type II: fail to reject a false H0, probability β. Power = 1 − β.
Using α instead of α ÷ 2 per tail in a two-tailed test.
Students read the critical value from the wrong column of the table.
Fix: For a two-tailed test at 5%, each tail holds 2.5%. Check the tails first, then read the table.
Reading the p-value as the probability that H0 is true.
The wording sounds like a probability about the hypothesis.
Fix: The p-value is the probability of a result at least this extreme if H0 is true. Compare it with α to decide.
Thinking a lower α is always better.
Students focus only on avoiding Type I error.
Fix: With a fixed sample size, lowering α raises β and cuts power. A larger sample reduces both errors.

Worked examples

Example 1

An analyst tests whether the mean monthly return of a global equity fund is different from 0.50%. A sample of 36 months gives a mean of 0.80% and a standard deviation of 1.20%. The test uses a 5% significance level, and the critical t-values for 35 degrees of freedom are ±2.030. What is the correct decision? A. Fail to reject H0, t = 1.50. B. Reject H0, t = 1.50. C. Reject H0, t = 3.00.

Show the solution

H0: μ = 0.50%. Ha: μ ≠ 0.50%. This is a two-tailed test.
Standard error = s ÷ √n = 1.20 ÷ √36 = 1.20 ÷ 6 = 0.20%.
t = (0.80 − 0.50) ÷ 0.20 = 0.30 ÷ 0.20 = 1.50.
Critical values are ±2.030. Since 1.50 lies between −2.030 and +2.030, it is not in the rejection region.
Fail to reject H0. Option B is wrong on the decision and option C is wrong on the statistic.

Answer: A. Fail to reject H0, t = 1.50.

Example 2

A researcher tests H0: a trading strategy's mean excess return ≤ 0 against Ha: mean excess return > 0. The test returns a p-value of 0.03 at a significance level of 5%. Which statement is correct? A. Fail to reject H0, because p-value is below 0.05. B. Reject H0; if H0 is in fact true, this decision is a Type I error. C. Reject H0; the probability that H0 is true is 3%.

Show the solution

The p-value is 0.03 and α is 0.05. Since 0.03 < 0.05, reject H0.
Option A says fail to reject, which contradicts the decision rule, so it is wrong.
Option C misreads the p-value. It is not the probability that H0 is true, so it is wrong.
Option B states the decision correctly and notes that rejecting a true H0 is a Type I error, which has probability α.

Answer: B. Reject H0; if H0 is in fact true, this decision is a Type I error.

Exam tips

Read Ha first. Most setup questions are solved by spotting ≠, > or <.
Expect options that say 'accept H0' or interpret the p-value as the probability H0 is true. Eliminate them at once.
For error questions, translate: Type I is a false alarm, Type II is a missed detection, power is the chance of catching a real effect.
If a question asks how to reduce both errors at once, the answer is a larger sample size.
In a two-tailed test, check whether the hypothesized value sits outside the confidence interval as a fast cross-check.

Practice questions from Estimation and Hypothesis Testing

Hypothesis Testing Framework in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Hypothesis Testing Framework: frequently asked questions

What are the steps of hypothesis testing in CFA Level I?

State H0 and Ha, choose the test statistic, set the significance level, find the decision rule, compute the statistic from the sample, then reject or fail to reject H0. Finish by stating what the decision means for the claim.

What is the difference between Type I and Type II error?

A Type I error is rejecting a true null hypothesis, and its probability is α. A Type II error is failing to reject a false null hypothesis, and its probability is β. Power equals 1 − β.

How do I compare the p-value with the significance level?

Reject H0 if the p-value is less than α. If the p-value is equal to or greater than α, fail to reject H0. The p-value is the smallest significance level at which H0 would be rejected.

How do I decide between a one-tailed and two-tailed test?

Look at Ha. If it uses ≠, run a two-tailed test and split α between the tails. If it uses > or <, run a one-tailed test and put all of α in the tail that Ha points to.

What is the power of a test?

Power is the probability of rejecting H0 when H0 is false, equal to 1 − β. It rises with a larger sample size and with a higher significance level, though a higher α also raises the chance of a Type I error.