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FRM Part I · FRM Exam Part I

Hypothesis Testing for FRM Part I: Chapter Guide

Hypothesis testing checks whether sample data supports a claim about a population. You state a null and an alternative, compute a test statistic, compare it with a critical value or p-value, and decide whether to reject the null. In FRM questions, most marks come from clean calculation and correct interpretation.

What this chapter covers

This chapter teaches you how to judge a claim using sample evidence. You set up a null hypothesis (H₀) and an alternative hypothesis (H₁), compute a test statistic, and compare it with a critical value or a p-value. The same logic applies whether you test a mean, a regression coefficient, a variance or a model's exceedance count.

The chapter has two layers. The first is mechanics: standard error, confidence intervals, t and z statistics, chi-square and F statistics. The second is interpretation: what a p-value does and does not mean, what Type I and Type II errors cost, and why a failure to reject is not proof that H₀ is true.

It sits at the centre of the Quantitative Analysis topic. It builds on probability, distributions and sampling, and it feeds straight into linear regression, where you test coefficients and the overall model. It also feeds Valuation and Risk Models, where VaR backtesting asks whether the number of exceptions is consistent with the stated confidence level. Chebyshev's inequality gives a distribution-free bound that helps when normality is doubtful.

Hypothesis testing is a tool you use across Quantitative Analysis and risk-model validation, so a weak grasp costs you marks in several chapters, not just one. The questions are calculation-heavy but short: a standard error, a t-statistic, a decision. If you learn the pattern once, you can solve these quickly and save time in a 100-question, 4-hour paper. Interpretation questions also reward precise wording, so careful study pays off twice.

Hypothesis Testing: topics in the order to study them

  1. 1Hypothesis Testing Framework and StepsStart here because every later test follows the same sequence: hypotheses, statistic, decision rule, conclusion.
  2. 2Confidence Intervals and Standard ErrorStandard error is the denominator of every test statistic, and intervals give a second route to the same decision.
  3. 3t-Test and z-Test for Means and Regression CoefficientsThis applies the framework to the most tested case and links directly to regression.
  4. 4p-Values, Type I and Type II Errors, and PowerOnce you can compute statistics, you can learn to interpret results and understand the cost of wrong decisions.
  5. 5Chi-Square and F-Tests for VarianceThese tests use new distributions and are asymmetric, so learn them after the familiar t and z logic is solid.
  6. 6Chebyshev's Inequality and Backtesting ApplicationsFinish with applications that combine the ideas and connect to VaR validation in later chapters.

How to prepare Hypothesis Testing

Treat this chapter as one procedure with different statistics plugged in. Practise the procedure until it is automatic, then add the interpretation rules.

  1. Write the five-step routine on one page: state H₀ and H₁, choose the test, compute the statistic, find the critical value or p-value, conclude. Use it on every question.
  2. Drill standard error until it is instant: sample standard deviation ÷ √n for a mean. Then build intervals as estimate ± critical value × standard error.
  3. Practise t-statistics as (estimate − hypothesised value) ÷ standard error, for sample means and for regression coefficients. Note the degrees of freedom each time.
  4. Learn the decision rules for one-tailed and two-tailed tests, and check that the p-value route and the critical-value route agree on the same problem.
  5. Memorise which statistic belongs to which question: chi-square for a single variance, F for comparing two variances or testing a regression as a whole.
  6. Work backtesting and Chebyshev questions last, then do mixed timed sets so you can pick the right test from the wording alone.
  7. Keep a financial calculator or the permitted tools ready for square roots and arithmetic, and practise getting through a full calculation in a couple of minutes.

Common mistakes in Hypothesis Testing

  • Stating the conclusion as 'accept H₀' or 'H₀ is proven'.

    Fix: Use 'reject' or 'fail to reject'. Failing to reject only means the evidence was not strong enough.

  • Reading the p-value as the probability that the null is true.

    Fix: Say it as: assuming H₀ is true, the chance of a result this extreme or more. Small p-value means the data is unlikely under H₀.

  • Dividing by s instead of s ÷ √n when testing a mean.

    Fix: Always compute the standard error as a separate line before the test statistic.

  • Mixing up one-tailed and two-tailed critical values.

    Fix: Read H₁ first. 'Different from' is two-tailed; 'greater than' or 'less than' is one-tailed.

  • Confusing Type I and Type II errors, or confusing power with significance.

    Fix: Type I = reject a true null, probability α. Type II = miss a false null. Power = 1 − Type II probability.

  • Using the wrong distribution for variance tests, such as a t-test for a variance.

    Fix: Match the parameter to the statistic: mean or coefficient with t or z, one variance with chi-square, two variances with F.

Last-day revision: Hypothesis Testing

  • H₀ is the claim you try to reject; failing to reject does not prove it true.
  • Standard error of a sample mean = s ÷ √n.
  • Test statistic = (estimate − hypothesised value) ÷ standard error.
  • Confidence interval = estimate ± critical value × standard error.
  • Reject H₀ when |statistic| exceeds the critical value, or when the p-value is below the significance level.
  • The p-value is the probability of a result at least this extreme if H₀ is true; it is not the probability H₀ is true.
  • Type I error: rejecting a true H₀; its probability is the significance level α.
  • Type II error: not rejecting a false H₀; power = 1 − probability of Type II error.
  • Lowering α reduces Type I errors but raises Type II errors for a fixed sample size.
  • A larger sample reduces standard error and raises power.
  • Chi-square tests one variance; F tests the ratio of two variances; both are non-negative and skewed.
  • Chebyshev: at most 1 ÷ k² of observations lie more than k standard deviations from the mean, for any distribution with finite variance and k > 1.

Hypothesis Testing practice questions

Hypothesis Testing 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: frequently asked questions

How much of the FRM Part I exam is hypothesis testing?

GARP publishes the topic structure but you should not rely on a fixed number of questions for one chapter. Hypothesis testing sits in Quantitative Analysis and its ideas appear in regression and risk-model questions too. Prepare it as a core skill.

When do I use a z-test and when a t-test?

Use z when the population standard deviation is known or the distribution of the statistic is treated as normal. Use t when you estimate the standard deviation from the sample, with n − 1 degrees of freedom for a mean. With large samples the two give very similar answers.

Do I need to memorise critical values?

Know the common normal values, such as about 1.645 for a one-tailed 5% test and 1.96 for a two-tailed 5% test. Questions usually supply other table values, so focus on knowing how to use them.

How does hypothesis testing connect to VaR backtesting?

Backtesting asks whether the number of days losses exceeded VaR is consistent with the stated confidence level. That is a hypothesis test on the exception rate. Too many or too few exceptions can lead you to reject the model.