Hypothesis Testing

Quantitative Methods. Worth 6 to 9 percent of the exam. One session: the lesson, the rules, the method, then the questions.

Quantitative MethodsHypothesis Testing
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The lesson

Runtime 14 minutes 9 seconds, measured from the published video.

The reading

Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.

The exam wants you to formulate a null and alternative hypothesis and identify one-tailed versus two-tailed tests, calculate the appropriate test statistic (z or t) and compare it to the correct critical value, interpret a p-value correctly, and describe the trade-off between Type I and Type II error. Nearly every calculation question runs the same five-step sequence.

Every hypothesis test on this exam runs through the same five steps in the same order: state the null hypothesis, H0 (always a statement of no effect or equality) and the alternative, Ha; choose a significance level; compute a test statistic; find the matching critical value or p-value; and reach a conclusion. The conclusion is always phrased as 'reject H0' or 'fail to reject H0,' never 'accept H0.' Failing to reject the null is not proof that it is true, only that this sample's evidence was not strong enough to overturn it, the same way a not-guilty verdict is not a certificate of innocence.

The choice between a z-statistic and a t-statistic turns on one fact only: whether the population standard deviation is actually known. If the problem states sigma, the population standard deviation, use z. If it gives you s, a standard deviation computed from the sample itself, use t, with n minus 1 degrees of freedom, regardless of how large the sample is. A large sample does not upgrade a t-test to a z-test; that shortcut belongs to a different context and is not the rule this exam tests. Both statistics are built the same way: the sample mean minus the hypothesized value, divided by the standard error, which is the standard deviation divided by the square root of n. Dividing by the standard deviation alone instead of the standard error is the most common arithmetic slip on this topic.

Whether a test is one-tailed or two-tailed changes which critical value applies. A two-tailed test, used when the alternative hypothesis simply claims inequality, splits the chosen significance level across both tails of the distribution; at 5 percent significance that gives a critical z of 1.96. A one-tailed test, used when the alternative claims a specific direction, puts the whole significance level in one tail, giving a smaller critical value of 1.645 at the same 5 percent level. Reading a one-tailed critical value off a two-tailed problem, or the reverse, is a common and entirely avoidable error.

The p-value method reaches the same conclusion a different way: it is the probability of observing a test statistic this extreme, or more extreme, if the null hypothesis were actually true. A p-value of 0.03 does not mean there is a 3 percent chance the null hypothesis is true; it means that, assuming the null is true, a result this far from expectation would happen only 3 percent of the time, which is unlikely enough to reject the null at a 5 percent significance level. The rule is simple: reject H0 when the p-value is smaller than the chosen significance level.

Two kinds of mistake are possible in any test, and they trade off against each other. A Type I error rejects a null hypothesis that was actually true, a false alarm, with probability equal to the significance level itself. A Type II error fails to reject a null hypothesis that was actually false, a missed detection, with probability called beta; the test's power, its ability to correctly detect a real effect, is 1 minus beta. Making the significance level stricter lowers the false-alarm rate but raises the missed-detection rate for a fixed sample size; the only way to reduce both at once is a larger sample.

A two-tailed rejection region under the null hypothesis reject H0 reject H0 fail to reject H0 -critical value +critical value
A two-tailed test splits the risk of a wrong rejection into both tails. The test statistic has to clear a critical value on either side before the null hypothesis is rejected.

Worked in full

An analyst tests whether the mean daily return of a fund differs from zero, using 25 observations. The sample mean is 0.42 percent and the sample standard deviation is 1.10 percent. At 5 percent significance (two-tailed), should the analyst reject the null hypothesis that the true mean return is zero? The problem gives s, not sigma, so this is a t-test with n - 1 = 24 degrees of freedom. Standard error = 0.0110 / sqrt(25) = 0.0110 / 5 = 0.0022. t = (0.0042 - 0) / 0.0022 = 1.91. The two-tailed 5 percent critical t-value at 24 degrees of freedom is 2.064. Since 1.91 is less than 2.064, the analyst fails to reject the null hypothesis: this sample does not provide strong enough evidence that the true mean daily return differs from zero.

The same problem, one step removed

Same data: n = 25, sample mean = 0.42 percent, sample standard deviation = 1.10 percent, testing against a hypothesized mean of zero at 5 percent significance, two-tailed. Build the standard error and the t-statistic yourself, then compare it to the two-tailed 5 percent critical value at 24 degrees of freedom (2.064).

The trap

An answer choice that says the null hypothesis is 'accepted,' or that interprets a p-value as the probability the null hypothesis is true, is wrong regardless of the arithmetic around it; both are language the exam treats as automatically disqualifying.

What this unit turns on

Read these before the questions, not after them. Everything here traces to this module's own lesson and to the 2026 outline.

Fail to reject, never accept

CFA language never says the null hypothesis is accepted. Failing to reject H0 means the evidence was not strong enough to reject it at the chosen significance level, not that H0 has been proven true. Any answer choice using the word accept for the null hypothesis is wrong on sight.

Type I is a false alarm; Type II is a missed detection

A Type I error rejects a null hypothesis that is actually true; its probability equals the chosen significance level, alpha. A Type II error fails to reject a null hypothesis that is actually false; its probability is called beta. The power of a test is 1 minus beta, the probability of correctly rejecting a false null.

Lowering alpha reduces Type I error but raises Type II error, for a fixed sample size

Making the significance threshold stricter makes rejection harder in every direction, so real effects are missed more often even as false alarms fall. The only way to reduce both error types together is to increase the sample size, which is not a lever available inside a single exam question's given data.

Known population standard deviation means z; sample standard deviation means t

The decision between a z-test and a t-test turns on whether the population standard deviation, sigma, is actually known, not on whether the sample is large. If the problem supplies s, the sample standard deviation, the test statistic is a t-statistic with n - 1 degrees of freedom, regardless of sample size; sigma given in the problem is the only trigger for a z-statistic.

A one-tailed test and a two-tailed test at the same significance level use different critical values

A two-tailed 5% test splits alpha into 2.5% in each tail, giving a critical z of 1.96; a one-tailed 5% test puts the full 5% in one tail, giving a critical z of 1.645. Reading a one-tailed critical value off a two-tailed row, or the reverse, is the single most common lookup error on this topic.

A single variance is tested with chi-square; two variances against each other are tested with an F-test

The chi-square statistic, (n-1) times sample variance divided by the hypothesized variance, tests whether one population's variance equals a stated constant. The F-statistic, the ratio of two sample variances with the larger variance conventionally placed in the numerator, tests whether two populations share the same variance. They answer different questions and are never interchangeable.

Parametric tests assume a distribution; nonparametric tests do not

A t-test, z-test, chi-square test, and F-test all assume the underlying population follows a specified distribution, normal in most CFA applications, and use its parameters (mean, variance) directly. A nonparametric test is used when that distributional assumption cannot reasonably be made, when the data is ranked rather than measured, or when the hypothesis concerns something other than a parameter, such as whether two variables are independent.

The trick

Greek letter, z; Roman letter, t

If the problem states sigma, the population standard deviation, use z. If it states s, the sample standard deviation, use t. This single check settles the z-versus-t decision faster than any sample-size rule of thumb.

1.645 and 1.96, memorized cold

One-tailed 5% significance: z = 1.645. Two-tailed 5% significance: z = 1.96. One-tailed 1%: z = 2.326. Two-tailed 1%: z = 2.576. A two-tailed test always has the larger critical value of the pair at the same significance level, because the same alpha is split across two tails instead of concentrated in one.

F has two legs, chi-square has one curve

F-test compares two variances against each other; chi-square tests one variance against a stated number. The larger sample variance always goes in the F-statistic's numerator, by convention, so F is never below 1.

Type I is convicting the innocent; Type II is acquitting the guilty

A courtroom analogy for the two error types: Type I error wrongly rejects a true null, like convicting an innocent defendant. Type II error wrongly fails to reject a false null, like acquitting a guilty one.

The method

The order to work a question of this type in, every time, before you touch the numbers.

  1. State the null and alternative hypotheses precisely, and note whether the test is one-tailed (a direction is claimed) or two-tailed (only inequality is claimed).
  2. Decide the test statistic: sigma known means z; s given means t; a single variance against a constant means chi-square; two variances against each other means F.
  3. Compute the test statistic, always dividing by the standard error (sample standard deviation divided by the square root of n), not by the standard deviation alone.
  4. Compare the test statistic to the correct critical value for the stated significance level and tail count, or compare the p-value to the significance level directly.
  5. State the conclusion using CFA's own language: reject H0, or fail to reject H0, never accept H0.
  6. [BA II Plus: the calculator does not compute critical values or p-values directly; enter the sample data with 2ND DATA and 2ND STAT to retrieve x-bar and Sx, then build the test statistic by hand as (x-bar minus the hypothesized value) divided by (Sx divided by the square root of n)]

Two worked examples, then you are on your own

The first is worked in full. The second gives you the setup and stops. After that the questions give you nothing, which is the point: the help fades on purpose, so the last thing you practise is the thing the exam actually asks of you.

Worked in full

An analyst wants to test whether the mean daily return of a portfolio is different from zero. She formulates H0: μ = 0 versus Ha: μ ≠ 0. With a sample of 36 daily returns, a sample mean of 0.15%, and a sample standard deviation of 0.45%, the test statistic is closest to:

Answer A. The correct answer is 2.00.

Your turn, setup given

Using the data from the previous question (t-stat = 2.00, n=36, two-tailed test), at a 5% significance level, the analyst should most likely:

State the null and alternative hypotheses precisely, and note whether the test is one-tailed (a direction is claimed) or two-tailed (only inequality is claimed).

The practice run

Pick an answer, say how sure you are, then reveal. Being sure and wrong is the most useful thing that can happen in a session, so answer honestly: it sends the unit back to learning and puts it at the front of your revision queue.

Question 1Exam level

A hypothesis test has a significance level of 5%. The probability of a Type II error is 20%. The power of the test is closest to:

How sure are you?

Correct: B. The correct answer is 80%.
A. Choosing 5% might tempt you if you confuse the significance level with the power of the test, but remember the power of the test is actually the probability of correctly rejecting a false null hypothesis, which here is 1 minus the probability of a Type II error, or 80%.
C. Choosing 75% might tempt you if you mistakenly subtract the significance level from the power, but remember, the power of the test is actually calculated as 1 minus the probability of a Type II error, making 80% the accurate value.

Unit: hypothesis-testing

Question 2Exam level

A researcher reports a p-value of 0.03 for a hypothesis test conducted at the 5% significance level. Which of the following conclusions is most appropriate?

How sure are you?

Correct: B. The correct answer is Reject the null hypothesis.
A. The most common p-value misconception. Interpreting p-value as the probability H0 is true. The p-value is the probability of observing a test statistic as extreme as calculated, ASSUMING H0 is true. It is NOT the probability that H0 is true. This distinction is directly tested.
C. You might see a small number and confuses direction. Small p-value means strong evidence against H0. A small p-value (below α) is evidence AGAINST H0, not in favor of it. If p < α, reject H0.

Unit: hypothesis-testing

Question 3Exam level

Which of the following best describes a Type I error in hypothesis testing?

How sure are you?

Correct: A. The correct answer is Rejecting the null hypothesis when it is actually true.
B. You might mix up Type I and Type II. Type II is failing to reject a false H0. Option B describes Type II error (β), not Type I. Type I = false positive (rejecting true null). Type II = false negative (failing to reject false null).
C. Conceptual confusion. Setting low α reduces Type I error but increases Type II. Setting α low reduces Type I error probability; it is not a definition of Type I error itself.

Unit: hypothesis-testing

Question 4Exam level

An analyst tests whether the variance of annual returns for a mutual fund equals 0.04 (σ² = 0.04). Using a sample of 25 annual returns with sample variance of 0.06, the chi-square test statistic is closest to:

How sure are you?

Correct: A. The correct answer is 36.00.
B. Choosing 24.00 might tempt you if you incorrectly used the sample mean instead of the sample variance in your calculation, but the chi-square test statistic requires the use of sample variance, leading to the correct calculation that results in 36.00.
C. Choosing 15.00 might tempt you if you incorrectly used the sample mean instead of the sample variance in your chi-square test statistic calculation, but the chi-square statistic requires using the sample variance, leading to the correct calculation of 36.00.

Unit: hypothesis-testing

Question 5Exam level

A portfolio manager tests whether the mean return of Portfolio A is greater than zero, using a one-tailed test at the 1% significance level with 30 observations and unknown population variance. The critical t-value is closest to:

How sure are you?

Correct: B. The correct answer is 2.462.
A. Choosing 2.326 might tempt you if you mistakenly used the z-table instead of the t-table, as 2.326 is the critical value for a one-tailed z-test at the 1% significance level, but for small sample sizes like 30 with unknown population variance, you must use the t-distribution, which gives a higher critical value of 2.462.
C. Choosing 2.756 might tempt you if you mistakenly used the critical value for a two-tailed test instead of a one-tailed test at the 1% significance level, which would lead to selecting a higher critical t-value than necessary for this scenario.

Unit: hypothesis-testing

Question 6Exam level

Two independent samples are drawn from populations. Sample 1 has variance s₁² = 0.09 (n₁=21) and Sample 2 has variance s₂² = 0.04 (n₂=16). An F-test for equality of variances (H0: σ₁² = σ₂²) at 5% significance (two-tailed). The F-statistic is closest to:

How sure are you?

Correct: A. The correct answer is 2.25.
B. Choosing 0.44 might seem correct if you mistakenly calculated the F-statistic by dividing the smaller variance by the larger variance, but the F-statistic requires dividing the larger variance by the smaller variance, making 0.44 incorrect.
C. Choosing 1.50 might be tempting if you mistakenly averaged the two sample variances, but the F-statistic requires you to divide the larger variance by the smaller variance, making 1.50 incorrect in this context.

Unit: hypothesis-testing

Question 7Exam level

Which statement about the relationship between Type I and Type II errors is most accurate?

How sure are you?

Correct: A. The correct answer is Decreasing the significance level increases the probability of a Type II error.
B. You might think 'lower significance = more careful = fewer mistakes of all kinds'. Lowering α reduces Type I error but INCREASES Type II error. You cannot reduce both simultaneously without increasing sample size.
C. Confusion about direction: larger n actually reduces Type II error (increases power). Increasing sample size reduces Type II error (increases power = 1-β). It does NOT increase Type II error.

Unit: hypothesis-testing

Question 8Exam level

An analyst tests whether the population mean return equals 8% (H0: μ = 8%). The sample mean is 10%, sample std dev is 6%, and n=25. The population standard deviation is unknown. The analyst should use which test, and the test statistic is closest to:

How sure are you?

Correct: A. The correct answer is t-test; t = 1.67.
B. You might see n=25 which is not > 30, but applies z-test anyway; or confuses z and t. When σ is UNKNOWN (sample std dev given), use t-test regardless of sample size. n=25 makes this even clearer. Both the unknown σ rule and small-n rule point to t-test.
C. You might be tempted to choose t = 0.33 if you incorrectly calculate the test statistic by dividing the sample mean by the sample standard deviation instead of using the correct formula which subtracts the hypothesized mean from the sample mean and divides by the standard error, leading to the correct t-value of 1.67.

Unit: hypothesis-testing

Question 9Exam level

A one-tailed test is conducted at the 5% significance level. The null hypothesis H0: μ ≤ 10 and alternative Ha: μ > 10. If the calculated test statistic is 1.72 and the critical value is 1.699, the correct conclusion is most likely:

How sure are you?

Correct: B. The correct answer is Reject H0; there is sufficient evidence that μ > 10.
A. You might be uncomfortable rejecting when the margin is small; tries to apply judgment. Hypothesis testing is binary at the stated significance level. If t-stat > critical value, reject H0. There is no 'close call' adjustment.
C. You might use incorrect CFA language. 'accept H0'. CFA curriculum NEVER uses 'accept the null hypothesis.' The correct language is always 'fail to reject H0.' Option C is wrong on both logic and language.

Unit: hypothesis-testing

Question 10Exam level

Which test statistic is most appropriate to test whether the variance of a normally distributed population equals a specified value?

How sure are you?

Correct: C. The correct answer is Chi-square statistic.
A. You might be tempted to choose the t-statistic because it is commonly used for hypothesis testing involving means, but the t-statistic is designed to test hypotheses about the mean of a population, not its variance, unlike the chi-square statistic which is specifically used for variance testing.
B. You might be tempted to choose the F-statistic because it is used for comparing variances, but the F-statistic is specifically for comparing two sample variances, not for testing a single population variance against a specified value like the chi-square statistic does.

Unit: hypothesis-testing

Question 11Above the exam

An analyst tests whether a fund's mean monthly return differs from zero at the 5% significance level and fails to reject the null hypothesis. A colleague argues this proves the fund's true mean return IS zero. Combining the logic of hypothesis testing with the concept of a Type II error, the colleague's claim is most likely:

How sure are you?

Correct: B. Hypothesis testing never proves the null hypothesis true; failing to reject it simply means the sample evidence was not strong enough, at the chosen significance level, to conclude the null is false. This is exactly the situation a Type II error describes: failing to reject a null hypothesis that is, in reality, false. The test's design cannot rule out that possibility, it can only control the probability of the OTHER kind of error (Type I, rejecting a true null) via the significance level chosen.
A. Classical hypothesis testing is built around DISPROVING (rejecting) the null hypothesis with sufficient evidence; it is never designed to statistically PROVE the null true. 'Fail to reject' and 'proven true' are different statistical claims, and confusing them is one of the most common conceptual traps on this LOS.
C. Whether the test was one-tailed or two-tailed does not change the fundamental logic that failing to reject a null hypothesis never proves it true; that limitation applies to hypothesis tests generally, not to a specific tail-count choice.

Unit: hypothesis-testing

Question 12Above the exam

A researcher runs the same hypothesis test on 20 different, unrelated fund managers' returns, each at the 5% significance level, and finds that exactly 1 manager's result is statistically significant. Applying the definition of significance level together with the logic of multiple testing, the most likely correct interpretation is that:

How sure are you?

Correct: B. A 5% significance level means that, by construction, there is a 5% chance of a Type I error (a false positive) on any single test where the null is actually true. Running 20 independent tests at 5% each, the EXPECTED number of false positives from chance alone is 20 x 0.05 = 1. Finding exactly one 'significant' result out of 20 is therefore consistent with pure random noise and provides weak, not strong, evidence of genuine skill for that one manager.
A. A single significant result out of 20 tests at the 5% level is exactly what chance alone predicts (1 expected false positive), so treating it as strong evidence of real skill ignores the multiple-testing context the question sets up.
C. There is nothing statistically invalid about applying the same significance level to several independent tests; the issue is not test validity, it is that running many tests raises the chance of at least one false positive by chance alone, which is a separate consideration from whether any individual test was performed correctly.

Unit: hypothesis-testing