Lesson 2 of 5 6–9% of the exam

You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

By Daniil Tarakanov Updated Oct 16, 20262026 curriculum

Type I and Type II errors and the power of a test

A Type I error rejects a null that is true; its probability is the significance level α. A Type II error fails to reject a null that is false; its probability is β. Power, 1 − β, is the chance of correctly rejecting a false null. Lowering α raises β, unless you add data.

6 min cram, 30 min full read. Based on the 2026 CFA® Program Level I curriculum, Quantitative Methods, Learning Module 8. This is the 2026 version of the lesson. Sitting in February 2027 or later? Read the updated 2027 version.

Tailor this to you: take the self-assessment

Before this: How a hypothesis test works: null, critical value, p-value

On the exam

Explain hypothesis testing and its components, including statistical significance, Type I and Type II errors, and the power of a test
Construct hypothesis tests and determine their statistical significance, the associated Type I and Type II errors, and power of the test given a significance level

Explain Type I and Type II errors and the power of a test, and determine them for a test at a given significance level. Expect scenario questions that name the error, and 'what happens to β if α falls?'.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

What can go wrong with a decision?

A test uses a sample, so it can be wrong in two different ways. Either it rejects a null that is actually true, or it fails to reject a null that is actually false. The size of each risk is something you can state and, to a degree, control.

The four outcomes

Memorize

The four outcomes of a test

Decision on H₀compared with the truth

Reject H₀ Memorize
- H₀ true: Type I error Memorize false alarm; probability α
- H₀ false: correct Memorize power; probability 1 − β
Fail to reject H₀ Memorize
- H₀ true: correct Memorize probability 1 − α
- H₀ false: Type II error Memorize missed signal; probability β

A Type I error is rejecting a true null hypothesis: a false alarm. Its probability is the significance level, α. A Type II error is failing to reject a false null hypothesis: a missed signal. Its probability is β.

The power of a test is the probability of correctly rejecting a false null.

Power of a test Memorize

\text{Power} = 1 - \beta

Exam tip Power is 1 − β, never 1 − α. The quantity 1 − α is the confidence level.

How do you put numbers on power?

Use it

Power depends on the cut-off for rejection and on where the true mean really is. With a known σ, the cut-off for a one-tailed test of H₀: μ ≤ μ₀ is a fixed number of standard errors above μ₀.

Rejection cut-off for a sample mean (σ known) Use it

c = \mu_0 + z \times \frac{\sigma}{\sqrt{n}}

Power from the cut-off Use it

\text{Power} = P\!\left(Z > \frac{c - \mu_1}{\sigma / \sqrt{n}}\right)

Worked example Power of a test of a mean return

H₀: μ ≤ 0 against Hₐ: μ > 0. The population standard deviation is known to be 2%, and n = 25. The true mean is 1.0%.

Standard error = 2 ÷ √25 = 0.4%.
Cut-off at α = 5%: 1.645 × 0.4 = 0.658%. Reject H₀ if the sample mean is higher.
If the true mean is 1.0%, the sample mean is centered on 1.0%, so power = P(X̄ > 0.658) = P(Z > (0.658 − 1.0) ÷ 0.4) = P(Z > −0.855) = 0.80.
So β = 0.20: one time in five the test misses a real 1.0% mean.

Now tighten α to 1% with the same n. The cut-off becomes 2.326 × 0.4 = 0.930%, and power falls to 0.57. Fewer false alarms, more missed signals.

Raise n to 100 at α = 5%. The standard error halves to 0.2%, the cut-off falls to 0.329%, and power rises to about 1.00. More data is the only way to cut both errors.

Steps 1 and 3 on your calculator

2÷25√x= 0.4
(0.658−1)÷0.4= −0.855, then read the z-table

√x square root

2ENTER25g√x÷ 0.4
0.658ENTER1−0.4÷ −0.855, then read the z-table

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Rearrange itRejection cut-off for a sample mean: solve for any one

The question asks for

For H₀: μ ≤ 0, σ is known to be 2.00% and n = 25. The critical value is z = 1.645. The sample mean above which H₀ is rejected is closest to:

Critical value, zPopulation standard deviation, σ%Sample size, n

c = μ_{0} + z \times \frac{σ}{n} (μ_{0} = 0)

1.645 × 2.00% ÷ √25 = 0.658%

Cut-off for the sample mean0.66%

On your calculator

1.645×2÷25√x=

√x square root

1.645ENTER2×25g√x÷

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

InteractiveTrade false alarms for missed signals

Try this 0 of 3

Keep n = 25 and μ₁ = 1.0%, and tighten α to 1%. Read the power.
Back at α = 5% and μ₁ = 1.0%, raise n until power is at least 0.900.
Slide the true mean down to 0 and read the power.

Drag the Cut-off for rejecting H₀, tap the chart, or focus the handle and use the arrow keys.

If H₀ is true (μ = 0)
If μ = 1.0%
α 0.050
β 0.196
Power 0.804

Red: α, rejecting a true H₀. Amber: β, missing a real effect. Green hatch: power.

Power (1 − β)0.804

Type I error α0.050

Type II error β0.196

The test (standard deviation known: 2%)

Significance level α5%

Sample size n25

The truth

True mean μ₁1.0%

Show working

Standard error = σ ÷ √n = 2 ÷ √25 = 0.4%.

Cut-off = z(α) × SE = 1.645 × 0.4 = 0.658%. Reject H₀ when the sample mean is above it. α = P(sample mean > cut-off | μ = 0) = 0.050.

If the true mean is 1.0%: z = (0.658 − 1.0) ÷ 0.4 = −0.855.

Power = P(Z > −0.855) = 0.804; β = 1 − power = 0.196.

Neither calculator has N(z): look the z up in the table. The keystrokes give the cut-off and z.

Lower α pushes the cut-off right: fewer false alarms, more missed signals. A bigger n narrows both curves and cuts both errors.

Cut-off and z under the true mean

2÷25√x=STO1SE 0.4
×1.645=cut-off 0.658
−1=÷RCL1=z = −0.855

√x square root · STO store · RCL recall a stored number

2ENTER25g√x÷STO1SE 0.4
1.645×cut-off 0.658
1−RCL1÷z = −0.855

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root · STO store · RCL recall a stored number

Display: −0.855

Move the cut-off and you trade false alarms for missed signals; only more data shrinks both.

Drag the cut-off line, or change α, n and the true mean, and watch the shaded areas for α, β and power change.

What moves the risks?

What moves the error risks and the power

Change one thing

Lower α (same n) Memorize fewer false alarms; β rises, power falls
Higher α (same n) Memorize β falls, power rises; more false alarms
Larger sample, n Memorize standard error shrinks; β falls and power rises at the same α
True mean farther from H₀ Use it easier to detect; power rises

Moving the cut-off only shifts risk from one error to the other: it is a trade. The significance level is a choice made before looking at the data, so it does not depend on n. The sample size changes the standard error, and that is the one lever that improves both β and power at once.

Go deeper Why 5% is the usual choice

The 5% level is a convention, not a law of nature. A researcher who fears false alarms more than missed signals (for example, before launching an expensive fund) may pick 1%. One who fears missing a real effect (for example, a safety screen) may accept 10%. State the level before the test; picking it afterwards to fit the result defeats the purpose.

Go deeper Many tests, many false alarms

Run 20 independent tests at 5% on data where nothing is going on, and you expect about one to reject by chance alone. This is why searching many strategies or signals until one looks significant (data snooping) overstates the evidence. It also explains why a pattern found by mining past prices should be re-tested on fresh data before anyone trusts it, a theme that returns in market efficiency.

Memorize this

MemorizeErrors and power

	$H_0$ true	$H_0$ false
Fail to reject $H_0$	correct ( $1-\alpha$ )	Type II error ( $\beta$ )
Reject $H_0$	Type I error ( $\alpha$ )	correct: power ( $1-\beta$ )

Memory hook Type I: you acted when you should not have (the alarm rings, no fire). Type II: you did nothing when you should have (fire, no alarm). The order matches the letters: α pairs with I, β with II.

Know it cold Significance level $= P(\text{Type I})$ . Power $= 1 - P(\text{Type II})$ . Not $1 - \alpha$ .

MemorizeTrade-offs

Change	Type I	Type II	Power
Lower $\alpha$ , same $n$	down	up	down
Larger $n$ , same $\alpha$	same	down	up

Memory hook Moving the cut-off only trades one error for the other. Only more data cuts both.

Know it cold To lower both errors, increase the sample size.

The four outcomes of a test

Decision on H₀compared with the truth

Reject H₀ Memorize
- H₀ true: Type I error Memorize false alarm; probability α
- H₀ false: correct Memorize power; probability 1 − β
Fail to reject H₀ Memorize
- H₀ true: correct Memorize probability 1 − α
- H₀ false: Type II error Memorize missed signal; probability β

What moves the error risks and the power

Change one thing

Lower α (same n) Memorize fewer false alarms; β rises, power falls
Higher α (same n) Memorize β falls, power rises; more false alarms
Larger sample, n Memorize standard error shrinks; β falls and power rises at the same α
True mean farther from H₀ Use it easier to detect; power rises

Rejection cut-off for a sample mean (σ known)

c = \mu_0 + z \times \dfrac{\sigma}{\sqrt{n}}

Quick example

H₀: μ ≤ 0, σ = 5%, n = 100, one-tailed 1% (z = 2.326). The cut-off is 2.326 × 5 ÷ √100 = 1.163%. Reject H₀ if the sample mean is higher.

On your calculator

2.326×5÷100√x= 1.163

√x square root

2.326ENTER5×100g√x÷ 1.163

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Power of a test

\text{Power} = 1 - \beta

Quick example

A test misses a false null 20% of the time (β = 0.20). Power = 1 − 0.20 = 0.80.

Power from the cut-off

\text{Power} = P\!\left(Z > \dfrac{c - \mu_1}{\sigma / \sqrt{n}}\right)

Quick example

Cut-off 0.98%, true mean 1.2%, standard error 0.5%. z = (0.98 − 1.2) ÷ 0.5 = −0.44, and P(Z > −0.44) = 0.67.

On your calculator

(0.98−1.2)÷0.5= −0.44

0.98ENTER1.2−0.5÷ −0.44

ENTER enter (on the HP, puts the number on the stack)

How it is asked

Name the error from a scenario

The analyst rejects the null that alpha is zero, but alpha is zero. Error?

Rejected a true null: Type I. Trap: Type II.
Effect of a lower significance level

Sample size fixed; α goes from 5% to 1%. What happens?

Type II probability rises, power falls. Trap: "power increases".
Define power

Power is the probability of…

Rejecting a false null (1 − β). Traps: rejecting a true null (that is α), failing to reject a false null (that is β).
Find a rejection cut-off

σ = 3%, n = 36, one-tailed 5%. Cut-off for the sample mean?

1.645 × 3 ÷ √36 = 0.8225%. Trap: using σ without √n.

Exam traps

Writing power = 1 − α. Power is 1 − β; 1 − α is the probability of correctly not rejecting a true null.
Saying a lower significance level raises power. With the same sample size it lowers power, because β rises.
Mixing up which error is 'rejecting a true null'. That is Type I. Failing to reject a false null is Type II.
Using σ instead of the standard error σ ÷ √n when finding a cut-off for a sample mean.

Checkpoint

4 exam-style questions. About 90 seconds each.

1An analyst rejects the null hypothesis that a manager's alpha is zero, when in truth the alpha is zero. This is a:
2Holding sample size constant, reducing the significance level from 5% to 1% will most likely:
3The power of a test is best described as the probability of:
4A population has a known standard deviation of 3%. An analyst tests H₀: μ ≤ 0 against Hₐ: μ > 0 with 36 observations at the 5% level (critical z = 1.645). H₀ is rejected if the sample mean exceeds a value closest to:

Key takeaways

Type I: reject a true H₀ (probability α). Type II: fail to reject a false H₀ (probability β).
Power = 1 − β: the chance of correctly rejecting a false null.
Lower α with the same n means higher β. Larger n lowers both errors.
Type I is a false alarm; Type II is a missed signal.

Rejection cut-off for a sample mean (σ known)

c = \mu_0 + z \times \dfrac{\sigma}{\sqrt{n}}

Quick example

H₀: μ ≤ 0, σ = 5%, n = 100, one-tailed 1% (z = 2.326). The cut-off is 2.326 × 5 ÷ √100 = 1.163%. Reject H₀ if the sample mean is higher.

On your calculator

2.326×5÷100√x= 1.163

√x square root

2.326ENTER5×100g√x÷ 1.163

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Power of a test

\text{Power} = 1 - \beta

Quick example

A test misses a false null 20% of the time (β = 0.20). Power = 1 − 0.20 = 0.80.

Power from the cut-off

\text{Power} = P\!\left(Z > \dfrac{c - \mu_1}{\sigma / \sqrt{n}}\right)

Quick example

Cut-off 0.98%, true mean 1.2%, standard error 0.5%. z = (0.98 − 1.2) ÷ 0.5 = −0.44, and P(Z > −0.44) = 0.67.

On your calculator

(0.98−1.2)÷0.5= −0.44

0.98ENTER1.2−0.5÷ −0.44

ENTER enter (on the HP, puts the number on the stack)

Next lesson t-tests of a mean, a difference in means and paired differences

Questions about this lesson?

Where this comes back

t-tests of a mean, a difference in means and paired differences Hypothesis Testing
Market Efficiency Equity Investments

Questions candidates ask

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

A Type I error rejects a null hypothesis that is actually true: a false alarm, with probability α. A Type II error fails to reject a null that is actually false: a missed signal, with probability β. Reducing one usually raises the other unless the sample size grows.

What is the power of a test?

Power is the probability that the test correctly rejects a false null hypothesis, equal to 1 − β. A powerful test rarely misses a real effect. Power rises with a larger sample, a higher significance level, and a true value farther from the null.

How does sample size affect Type I and Type II errors?

The Type I error probability is whatever significance level you choose, so sample size does not change it. A larger sample shrinks the standard error, which lowers the Type II error probability and raises power at the same α. It is the only way to reduce both risks.

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