Quantitative Methods, LOS weight share 0.8 percent of the 365 Level I learning outcomes.
The exam never asks whether the null hypothesis is true, only whether the evidence is strong enough to reject it, and every wrong answer choice that says accept the null is disqualified before a single number is checked.
Answer these three first. Getting them wrong now is normal, and it helps the lesson stick. Reveal the answers when you're done, then read on.
1. A researcher reports a p-value of 0.03 for a test conducted at the 5% significance level. The most appropriate conclusion is:
2. A hypothesis test is conducted with a sample standard deviation but no known population standard deviation, and a sample size of 25. The appropriate test statistic is:
3. A test's significance level is lowered from 5% to 1%, holding sample size fixed. The most likely effect is:
Runtime 14 minutes 9 seconds, measured from the published video.
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.
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.
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).
n = 25, sample mean = 0.42%, sample s = 1.10%, H0: mu = 0, two-tailed, 5% significance. Reject or fail to reject?
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.
Verbatim from the 2026 CFA Level I topic outline. Every practice question and key rule below is tagged to one of these where the stem and explanation make the match clear.
Written from this module's own lesson and the 2026 CFA Level I topic outline, in teaching order, each tagged to the learning outcome it belongs to where that is clear.
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.
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.
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.
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 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.
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.
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.
Authored only where a key rule has an arbitrary number, list, or formula shape worth a memory device; a module with none of those has no tricks here, on purpose.
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.
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-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.
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.
Authored, ordered steps for answering this module's question types; a calculation module's calculator-dependent step ends with a bracketed BA II Plus keystroke sequence.
Condensed from the key rules and tricks above, nothing new. What you'd want on one index card the night before.
Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. 10 question(s) available for this unit.
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:
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Unit: hypothesis-testing
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:
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Unit: hypothesis-testing
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:
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Unit: hypothesis-testing
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?
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Unit: hypothesis-testing
Which of the following best describes a Type I error in hypothesis testing?
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Unit: hypothesis-testing
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:
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Unit: hypothesis-testing
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:
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Unit: hypothesis-testing
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:
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Unit: hypothesis-testing
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:
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Unit: hypothesis-testing
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:
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Unit: hypothesis-testing
Answer the questions above, then press the button.