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

Learning Module 8 of Quantitative Methods 6–9% of the exam

Hypothesis Testing

5 lessons. About 32 minutes in cram mode, 160 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary

Lessons

1
How a hypothesis test works: null, critical value, p-value
explain 8 min cram, 36 min full
2
Type I and Type II errors and the power of a test
explain 6 min cram, 30 min full
3
t-tests of a mean, a difference in means and paired differences
construct 8 min cram, 40 min full
4
Chi-square and F-tests of variance
construct 6 min cram, 30 min full
5
Parametric vs nonparametric tests
compare 4 min cram, 24 min full

What the exam expects you to do

Explain hypothesis testing and its components, including statistical significance, Type I and Type II errors, and the power of a test.How a hypothesis test works: null, critical value, p-value
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.How a hypothesis test works: null, critical value, p-value
Compare and contrast parametric and nonparametric tests, and describe situations where each is the more appropriate type of test.Parametric vs nonparametric tests

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

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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.

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

Known

Memorizez critical values

Significance level α	One-tailed	Two-tailed
10%	1.28	1.645
5%	1.645	1.96
1%	2.33	2.575

Memory hook Two-tailed at α is one-tailed at α ÷ 2. So the two-tailed 5% value (1.96) is the one-tailed 2.5% value, and 1.645 appears twice: one-tailed 5% and two-tailed 10%.

Know it cold One-tailed values are smaller than two-tailed values at the same α. Smaller α pushes the critical value farther out.

MemorizeDecision rules and wording

Item	Rule
Reject $H_0$ when	statistic beyond the critical value, or p-value less than $\alpha$
Equals sign	always in $H_0$ ( $=$ , $\le$ or $\ge$ )
Claim to prove	goes in $H_a$
Confidence level	$1 - \alpha$
p-value	smallest $\alpha$ at which $H_0$ is rejected
Wording	"reject" or "fail to reject", never "accept"

Memory hook The null is innocent until proven guilty: the evidence either convicts it or fails to convict it. It is never declared proven innocent.

Know it cold A p-value is not the probability that $H_0$ is true.

Key terms

hypothesis Know the idea
hypothesis testing Know the idea
null hypothesis Memorize
alternative hypothesis Memorize
test statistic Memorize
level of significance Memorize
confidence level Use it
critical values Memorize
statistically significant Use it
p-value Memorize
one-tailed test Memorize
two-tailed test Memorize

The six steps of a hypothesis test

Hypothesis testfrom a claim to a decision

1. State the hypotheses Memorize H₀ (with the equals sign) and Hₐ
2. Identify the test statistic Memorize and its distribution
3. Choose the significance level Memorize α, usually 5%
4. State the decision rule Memorize critical value(s)
5. Compute the statistic Memorize from the sample
6. Decide Memorize reject or fail to reject, then the economic decision

Which form of hypotheses?

The claim you want to showit goes in Hₐ; the equals sign stays in H₀

Two-tailed test Memorize the claim is "different from"
- H₀: μ = μ₀ versus Hₐ: μ ≠ μ₀ Memorize α split across both tails; z at 5%: ±1.96
One-tailed test, right Memorize the claim is "greater than"
- H₀: μ ≤ μ₀ versus Hₐ: μ > μ₀ Memorize all of α in the right tail; z at 5%: +1.645
One-tailed test, left Memorize the claim is "less than"
- H₀: μ ≥ μ₀ versus Hₐ: μ < μ₀ Memorize all of α in the left tail; z at 5%: −1.645

Two equivalent ways to decide

Reject H₀ when

Critical-value rule Memorize the test statistic is beyond the critical value
p-value rule Memorize the p-value is less than α

Formulas

Test statistic Memorize

\text{Test statistic} = \dfrac{\text{Sample statistic} - \text{Hypothesized value}}{\text{Standard error of the sample statistic}}

Quick example

A sample mean of 7.2% against a hypothesized 6%, with a standard error of 0.8%: (7.2 − 6) ÷ 0.8 = 1.50 standard errors above the null.

On your calculator

(7.2−6)÷0.8= 1.5

7.2ENTER6−0.8÷ 1.5

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

Test statistic for a single mean Memorize

t = \dfrac{\bar{X} - \mu_0}{s \,/\, \sqrt{n}},\quad df = n - 1

Quick example

16 returns average 1.5% with s = 3%. For H₀: μ = 0.5%, t = (1.5 − 0.5) ÷ (3 ÷ √16) = 1 ÷ 0.75 = 1.33, with 15 degrees of freedom.

On your calculator

(1.5−0.5)÷(3÷16√x)= 1.3333

√x square root

1.5ENTER0.5−3ENTER16g√x÷÷ 1.3333

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

Confidence level Use it

\text{Confidence level} = 1 - \alpha

Quick example

A 1% significance level is a 99% confidence level.

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

Known

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.

Key terms

Type I error Memorize
Type II error Memorize
power of a test 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

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

Formulas

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

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 Memorize

\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 Use it

\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)

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

Known

MemorizeThe three t-tests

Data	Statistic	df
One sample vs a value	$\dfrac{\bar{X}-\mu_0}{s/\sqrt{n}}$	$n-1$
Two independent samples	$\dfrac{\bar{X}_1-\bar{X}_2}{\sqrt{s_p^2/n_1+s_p^2/n_2}}$	$n_1+n_2-2$
Paired samples	$\dfrac{\bar{d}}{s_d/\sqrt{n}}$	$n-1$

Memory hook Every t is a difference over its standard error. Degrees of freedom are the observations minus the number of means you estimated: one mean, n − 1; two means, n₁ + n₂ − 2; one mean of differences, n − 1.

Know it cold Pooled variance: $s_p^2=\dfrac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$ . Samples linked by date or subject: paired, never pooled.

MemorizeDecision with a t-statistic

Hₐ	Reject when
$\ne$ (two-tailed)	$\|t\| >$ two-tailed critical value
$>$ (one-tailed)	$t >$ critical value
$<$ (one-tailed)	$t <$ −critical value

Memory hook Read the sign in Hₐ before opening the t-table.

Know it cold The critical value comes from the degrees of freedom, so find $df$ first.

Key terms

t-distribution Use it
degrees of freedom Memorize
test of a single mean Memorize
test of the difference in means Memorize
pooled variance Memorize
test of the mean of the differences Memorize
paired comparisons test Memorize
dependent samples Use it

Which t-test fits the data?

How were the data collected?this decides the standard error and the degrees of freedom

One sample against a value Memorize
- Test of a single mean Memorize t = (X̄ − μ₀) ÷ (s ÷ √n); df n − 1
Two independent samples Memorize unrelated observations
- Test of the difference in means Memorize pooled variance; df n₁ + n₂ − 2
Two dependent (paired) samples Memorize linked observations, such as the same months
- Paired comparisons test Memorize test the differences; t = d̄ ÷ (s_d ÷ √n); df n − 1

Formulas

Pooled variance Memorize

s_p^2 = \dfrac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}

Quick example

n₁ = 20, s₁ = 3%; n₂ = 15, s₂ = 2.5%. s²ₚ = (19 × 9 + 14 × 6.25) ÷ 33 = 258.5 ÷ 33 = 7.83 (%²).

On your calculator

(19×3x²)+(14×2.5x²)=÷33= 7.8333

x² square

19ENTER3ENTER××14ENTER2.5ENTER××+33÷ 7.8333

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

Test of the difference in means (independent samples, equal variances) Memorize

t = \dfrac{\bar{X}_1 - \bar{X}_2}{\sqrt{\dfrac{s_p^2}{n_1} + \dfrac{s_p^2}{n_2}}},\quad df = n_1 + n_2 - 2

Quick example

Means 8.0% and 6.5%, s²ₚ = 7.83, n₁ = 20, n₂ = 15. SE = √(7.83 × (1/20 + 1/15)) = 0.956, so t = 1.5 ÷ 0.956 = 1.57 with 33 degrees of freedom.

On your calculator

201/x+151/x=×7.8333=√xSTO1 0.956
(8−6.5)÷RCL1= 1.569

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

201/x151/x+7.8333×g√xSTO1 0.956
8ENTER6.5−RCL1÷ 1.569

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

Mean of the differences Memorize

\bar{d} = \dfrac{1}{n}\sum_{i=1}^{n} d_i,\quad d_i = X_{1i} - X_{2i}

Quick example

Four monthly differences of 0.4, −0.1, 0.6 and 0.5 (percentage points): (0.4 − 0.1 + 0.6 + 0.5) ÷ 4 = 0.35.

On your calculator

0.4−0.1+0.6+0.5=÷4= 0.35

0.4ENTER0.1−0.6+0.5+4÷ 0.35

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

Paired comparisons test Memorize

t = \dfrac{\bar{d} - \mu_{d0}}{s_d \,/\, \sqrt{n}},\quad df = n - 1

Quick example

25 paired months, d̄ = 0.8%, s_d = 1.5%, H₀: μ_d = 0. t = 0.8 ÷ (1.5 ÷ √25) = 0.8 ÷ 0.3 = 2.67 with 24 degrees of freedom.

On your calculator

0.8÷(1.5÷25√x)= 2.6667

√x square root

0.8ENTER1.5ENTER25g√x÷÷ 2.6667

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

Chi-square and F-tests of variance

Known

MemorizeVariance tests

	Chi-square	F
Tests	one variance	two variances
Statistic	$\dfrac{(n-1)s^2}{\sigma_0^2}$	$\dfrac{s_1^2}{s_2^2}$ (larger on top)
df	$n-1$	$n_1-1$ and $n_2-1$
Shape	non-negative, skewed right	non-negative, skewed right
Two-tailed critical values	lower and upper	upper only, at $\alpha/2$

Memory hook Variances are squares, so both distributions start at zero and have a long right tail. Putting the larger variance on top means the F-statistic is never below 1, so you only need the upper critical value.

Know it cold Square the standard deviations first. Use $n-1$ , not $n$ . Both tests assume normal populations.

Key terms

chi-square distribution Use it
test of a single variance Memorize
F-distribution Use it
test of the difference in variances Memorize

Which variance test?

What is being tested?squared numbers, so the distributions start at zero and lean right

One variance against a value Memorize
- Chi-square test Memorize χ² = (n − 1)s² ÷ σ₀²; df n − 1
Two variances against each other Memorize
- F-test Memorize F = larger s² ÷ smaller s²; df n₁ − 1 and n₂ − 1

Formulas

Chi-square test of a single variance Memorize

\chi^2 = \dfrac{(n-1)\,s^2}{\sigma_0^2},\quad df = n - 1

Quick example

16 monthly returns with s = 3.0%, tested against σ₀ = 2.0%. χ² = 15 × 9 ÷ 4 = 33.75, with 15 degrees of freedom.

On your calculator

15×3x²÷2x²= 33.75

x² square

15ENTER3ENTER××2ENTER×÷ 33.75

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

F-test of the equality of two variances Memorize

F = \dfrac{s_1^2}{s_2^2}\quad(s_1^2 \ge s_2^2),\quad df = (n_1 - 1,\ n_2 - 1)

Quick example

Standard deviations of 8% and 5%. F = 64 ÷ 25 = 2.56, larger variance on top.

On your calculator

8x²÷5x²= 2.56

x² square

8ENTER×5ENTER×÷ 2.56

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

Parametric vs nonparametric tests

Known

MemorizeParametric and nonparametric pairs

Question	Parametric	Nonparametric
Single mean	t-test (or z)	Wilcoxon signed-rank
Two independent means	t-test (pooled)	Mann–Whitney U (Wilcoxon rank-sum)
Paired differences	paired t-test	Wilcoxon signed-rank; sign test

Memory hook One signed-rank test serves two jobs (a single sample and paired differences). Two independent samples are the only ones that need the rank-sum / Mann–Whitney test.

Know it cold Nonparametric = few assumptions, but less power when the parametric assumptions do hold.

MemorizeFour situations for a nonparametric test

#	Situation
1	Distributional assumptions fail
2	Outliers
3	Ranks or ordinal data
4	Hypothesis is not about a parameter

Memory hook Break, Bump, Rank, Beyond: assumptions that break, outliers that bump the mean, data that are ranks, questions beyond parameters.

Know it cold If the assumptions are met, prefer the parametric test (more powerful).

Key terms

parametric test Use it
nonparametric test Use it
ranks Know the idea
Wilcoxon signed-rank test Memorize
Mann–Whitney U test Memorize
sign test Memorize

When to use a nonparametric test

Nonparametric testfew assumptions; ranks and signs

Assumptions fail Memorize small sample, clearly non-normal data
Outliers Memorize extremes would swamp a mean-based test
Ranked or ordinal data Memorize fund rankings, credit grades
Question is not about a parameter Memorize is this sequence random? does it follow this distribution?

Matching a parametric test with its nonparametric alternative

The question

Single mean Memorize
- Parametric: t-test (or z-test) Memorize
- Nonparametric: Wilcoxon signed-rank test Memorize
Difference between two independent means Memorize
- Parametric: t-test (pooled) Memorize
- Nonparametric: Mann–Whitney U test (Wilcoxon rank-sum test) Memorize
Mean of paired differences Memorize
- Parametric: paired comparisons t-test Memorize
- Nonparametric: Wilcoxon signed-rank test or sign test Memorize

Cheat sheet · PDF · 4 A4 pages

Hypothesis Testing on 4 pages

What to memorize, the concept maps, every formula with a worked example and keystrokes for the TI BA II Plus and HP 12C, how each idea is asked, and the traps. Print it, or keep it on your phone.

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