FRM Exam Part I · Hypothesis Testing

Hypothesis Testing Framework and Steps for FRM Part I

Updated 11 October 2026 · Fact-checked

Hypothesis testing checks whether sample evidence is strong enough to reject a stated claim, the null hypothesis. You state H0 and H1, choose a significance level, compute a test statistic, compare it with the critical value, and reject H0 only if the statistic falls in the rejection region.

Understand Hypothesis Testing Framework and Steps

A hypothesis test is a rule for deciding between two claims using sample data. The null hypothesis (H0) is the default claim, such as 'the mean daily return is zero' or 'the slope coefficient is zero'. The alternative hypothesis (H1) is what you accept if the data contradict H0. H0 always contains the equality sign (=, ≤ or ≥).

The test statistic measures how far your sample estimate is from the value in H0, in units of standard error. If that distance is large, the sample would be unlikely if H0 were true. The significance level (α) sets how unlikely is 'too unlikely'. It is the probability of rejecting H0 when H0 is true (a Type I error).

The form of H1 decides the tails. A two-tailed test uses H1 with 'not equal' (≠) and splits α between both tails. A one-tailed test uses H1 with '>' or '<' and puts all of α in one tail. The one-tailed critical value is closer to zero, so it is easier to reject in the stated direction.

The decision rule is simple. If the test statistic falls beyond the critical value, it is in the rejection region and you reject H0. Otherwise you fail to reject H0. You never 'accept' H0 or 'prove' it. Failing to reject only means the evidence was not strong enough.

In finance you might test whether a fund's alpha is zero, whether a VaR model's exception rate matches its confidence level, or whether a regression beta differs from 1. The framework is the same each time.

Key formulas to remember

Test statistic for a mean: t = (x̄ − μ0) ÷ (s ÷ √n); Use t with n − 1 degrees of freedom when the population variance is unknown. Use z when σ is known or n is large.
Standard error of the sample mean: SE = s ÷ √n; Divide the sample standard deviation by the square root of the sample size.
General test statistic: (estimate − hypothesized value) ÷ standard error of estimate; Works for means, regression coefficients and differences.
Decision rule, two-tailed: Reject H0 if |test statistic| > critical value; Critical values use α ÷ 2 in each tail. For z at 5%: ±1.96. At 1%: ±2.58.
Decision rule, one-tailed: Right tail: reject if statistic > critical value. Left tail: reject if statistic < −critical value; For z at 5%: 1.645. At 1%: 2.33.
Confidence interval link: x̄ ± critical value × SE; A two-tailed test at α rejects H0 when μ0 lies outside the (1 − α) confidence interval.

How to solve Hypothesis Testing Framework and Steps questions

Use the same sequence on any hypothesis testing question. Write each step down briefly so you do not mix up directions.

1State H0 and H1 from the wording. 'Differs from' means two-tailed. 'Greater than' or 'less than' means one-tailed.
2Note the significance level α and whether the question gives σ (use z) or only s (use t with n − 1 degrees of freedom).
3Compute the standard error, then the test statistic = (estimate − hypothesized value) ÷ SE.
4Find the critical value for α, the tail type and the distribution. For two-tailed tests, use α ÷ 2 per tail.
5Compare: reject H0 if the statistic is beyond the critical value in the direction of H1.
6State the conclusion in words: 'reject H0' or 'fail to reject H0', and link it back to the finance context.

Quickest way: Compare the statistic to memorised critical values

When to use it: Use when the question gives a sample mean, a standard error or a t-statistic and asks for the decision. This is the fastest route under time pressure.

Memorise z critical values: 1.645 and 1.96 (5% one-tailed and two-tailed), 2.33 and 2.58 (1%).
Compute (estimate − hypothesized) ÷ SE in one line.
Check the sign matches H1 for a one-tailed test. A large statistic in the wrong direction does not reject.
If the answer options differ only by 'reject' or 'fail to reject', decide using |statistic| against the critical value and eliminate the rest.

Common mistakes in Hypothesis Testing Framework and Steps

Saying 'accept H0' when the statistic is not significant
Everyday language treats the opposite of rejecting as accepting.
Fix: Always write 'fail to reject H0'. The test cannot prove H0 true.
Using the two-tailed critical value for a one-tailed test
Students memorise 1.96 and apply it everywhere.
Fix: Read H1 first. If it has > or <, use the one-tailed value (1.645 at 5%).
Putting the claim to be proven in H0
The claim seems like the natural starting point.
Fix: H0 holds the equality and the status quo. The effect you want evidence for goes in H1.
Rejecting in the wrong direction on a one-tailed test
Students compare absolute values only.
Fix: For a left-tailed test the statistic must be below the negative critical value. A large positive statistic does not reject.
Using z when only the sample standard deviation is known and n is small
z tables are better memorised than t tables.
Fix: With unknown σ, use t with n − 1 degrees of freedom. The t critical value is larger, especially for small n.
Confusing α with the probability that H0 is true
Both are described as probabilities about H0.
Fix: α is the probability of rejecting H0 given H0 is true (Type I error). It says nothing about the probability H0 is true.

Worked examples

Example 1

A risk analyst tests whether a fund's mean monthly excess return differs from zero. Sample of 36 months: mean 0.80%, standard deviation 3.0%. Use a 5% significance level. Critical values for the t-distribution are ±2.030 (35 degrees of freedom). What is the decision?

Show the solution

H0: μ = 0. H1: μ ≠ 0 (two-tailed).
SE = 3.0% ÷ √36 = 3.0% ÷ 6 = 0.50%.
t = (0.80% − 0) ÷ 0.50% = 1.60.
Critical values are ±2.030. Since |1.60| < 2.030, the statistic is not in the rejection region.

Answer: Fail to reject H0. There is not enough evidence at 5% that the mean excess return differs from zero.

Example 2

A bank claims its average daily trading loss is no more than USD 2.0 million. A sample of 100 days gives a mean loss of USD 2.4 million with a standard deviation of USD 2.0 million. Test at the 5% level (z critical value 1.645, one-tailed).

Show the solution

H0: μ ≤ 2.0 million. H1: μ > 2.0 million (right-tailed).
SE = 2.0 ÷ √100 = 2.0 ÷ 10 = 0.20 million.
z = (2.4 − 2.0) ÷ 0.20 = 2.00.
Compare with 1.645. Since 2.00 > 1.645 and the sign matches H1, the statistic is in the rejection region.

Answer: Reject H0. At the 5% level there is evidence that the average daily loss exceeds USD 2.0 million.

Exam tips

Read the wording for direction. 'Different from' is two-tailed. 'Exceeds' or 'is less than' is one-tailed.
Know the z values 1.645, 1.96, 2.33 and 2.58 by heart. Questions often provide t tables, but z values save time.
Wrong options are often built from the wrong tail count or from 'accept H0'. Check both before choosing.
If a question gives a confidence interval, you can test a two-tailed hypothesis by checking whether the hypothesized value lies inside it.
Questions may ask for the effect of changing α. A smaller α makes the critical value larger and rejection harder.

Practice questions from Hypothesis Testing

Hypothesis Testing Framework and Steps 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 and Steps: frequently asked questions

What are the steps of hypothesis testing in FRM Part I?

State H0 and H1, choose the significance level, compute the test statistic, find the critical value, and compare. Reject H0 if the statistic lies in the rejection region. Then state the conclusion in context.

What is the difference between a one-tailed and a two-tailed test?

A two-tailed test checks for a difference in either direction and splits α across both tails. A one-tailed test checks a single direction and places all of α in one tail. The one-tailed critical value is smaller in magnitude.

Why do we say 'fail to reject' instead of 'accept' the null?

A non-significant result means the sample did not give strong enough evidence against H0. It does not show that H0 is true. A larger sample or a different test might reject it.

How do I choose between a z-test and a t-test?

Use z when the population standard deviation is known or the sample is large. Use t with n − 1 degrees of freedom when only the sample standard deviation is available. The t critical value is larger for small samples.