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Set up the null and the alternative hypothesis

CFA® Level I · Quantitative Methods · Estimation and Hypothesis Testing · QM.7.a

The idea

The null hypothesis is the statement being tested, and it always contains the equality: it is the position held until the evidence is strong enough to abandon it. The alternative is what the analyst suspects and will conclude only if the null is rejected. A one-tailed test asks whether a parameter is greater, or smaller, than a value; a two-tailed test asks only whether it differs. The direction must be chosen before seeing the data, because picking the tail that suits the result turns a test into a formality. Failing to reject a null is not proof that it is true.

A practice question

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

Which hypothesis always contains the equality?

  1. The alternative hypothesisWhy not: The alternative carries the strict inequality. The equality has to sit with the null because the test statistic is computed against a specific value, and only an equality supplies one.
  2. The null hypothesis — correct
  3. Both contain itWhy not: They are mutually exclusive by construction — between them they must cover every possibility exactly once.

The null is the position assumed true, so it carries the equality and is what the test statistic is computed against. The alternative is the strict inequality.

A harder one

An analyst tests whether a fund's mean return differs from zero in either direction. What test is appropriate?

  1. A two-tailed test — correct
  2. A one-tailed test with the upper tailWhy not: That tests whether the mean is *greater* than zero, and would miss a fund reliably losing money.
  3. A one-tailed test with the lower tailWhy not: That tests whether the mean is *less* than zero. 'Differs' is directionless, so both tails must be live.

'Differs' is directionless, so rejection regions sit in both tails and the significance level is split between them. Testing only one tail would answer a narrower question.

The hardest tier

Why must the direction of a one-tailed test be chosen before examining the data?

  1. Because a two-tailed test is always more powerfulWhy not: It is less powerful against a correctly specified one-sided alternative, since the significance level is split across two tails. Power is not the reason.
  2. Because the test statistic changes with directionWhy not: The statistic is the same number either way. What changes is the critical value it is compared against, and which tail counts as rejection.
  3. Because choosing the tail after seeing the result inflates the true rate of false positives — correct

Picking whichever tail the data favours doubles the effective significance level while reporting the original one. The stated probability of a false positive then understates the real one.

In the app: 9 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.

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