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Formulate a suitable null hypothesis and alternative hypothesis and set the two apart.

FRM® Part I · Quantitative Analysis · Hypothesis Testing · FRM1.QA.6.a

The idea

A hypothesis test starts by stating two claims about a population parameter, never about the sample statistic. The null hypothesis H0 is the default position the test tries to reject. It always carries the equality, such as μ = 0, μ ≤ 0 or μ ≥ 0. The alternative hypothesis H1 is the complement and holds the strict inequality or the not-equal sign. Put the claim you want evidence for in the alternative, because only a rejection counts as evidence for the claim. Failing to reject only means the data are not strong enough at the chosen significance level, and it never proves the null true. Fix both hypotheses before looking at the data. The exam favours the choice with the equality in the null and the claim to be proved in the alternative.

A practice question

From the question bank, with the reason each wrong option is wrong.

An analyst wants to test whether the mean daily return of a trading strategy differs from zero. Which pair of hypotheses is correctly formulated?

  1. H0: mu not equal to 0 versus H1: mu = 0Why not: Puts the equality in the alternative; the null hypothesis must be the one containing the equality sign.
  2. H0: mu = 0 versus H1: mu not equal to 0 - correct
  3. H0: sample mean = 0 versus H1: sample mean not equal to 0Why not: States the hypotheses about the sample statistic, which is observed; hypotheses concern the population parameter mu.
  4. H0: mu > 0 versus H1: mu < 0Why not: Leaves the equality out of both hypotheses, so mu = 0 is covered by neither and the pair does not exhaust the possibilities.

Hypotheses are statements about the population parameter. The null contains the equality (mu = 0) and the alternative holds the not-equal sign because the question asks whether the mean differs from zero in either direction.

A harder one

A risk committee has set a volatility limit of 18 percent a year. The risk manager wants statistical evidence that a portfolio's true volatility is below the limit so that it can be approved. Which hypotheses should she use?

  1. H0: sigma >= 18 percent versus H1: sigma < 18 percent - correct
  2. H0: sigma < 18 percent versus H1: sigma >= 18 percentWhy not: Reverses the roles; the claim to be established must sit in the alternative so that rejecting the null is real evidence for it.
  3. H0: sigma = 18 percent versus H1: sigma not equal to 18 percentWhy not: Uses a two-sided alternative although the claim of interest is directional (below the limit).
  4. H0: sigma <= 18 percent versus H1: sigma > 18 percentWhy not: Tests whether volatility exceeds the limit, the opposite of the claim she wants to establish.

Rejection is the only conclusive outcome, so the claim to be supported (volatility below 18 percent) goes in H1 and the null carries the equality: H0: sigma >= 18 percent.

The hardest tier

A desk wants to demonstrate that the mean daily P&L of a hedged book is positive. Which set of hypotheses lets a rejection of the null support that claim?

  1. H0: mu <= 0 versus H1: mu > 0 - correct
  2. H0: mu >= 0 versus H1: mu < 0Why not: Puts the claim to be shown (not negative) in the null, so a failure to reject would wrongly be read as proof.
  3. H0: mu < 0 versus H1: mu >= 0Why not: Places the equality sign in the alternative; the null must contain the equality.
  4. H0: mu = 0 versus H1: mu < 0Why not: Tests for a loss, the opposite direction from the claim, so rejection would not support it.

The claim to be supported goes in the alternative and the null carries the equality. Rejecting H0: mu <= 0 gives evidence in favour of positive mean P&L.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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