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Tell one-sided tests from two-sided tests and decide which is appropriate in a given situation.

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

The idea

The alternative hypothesis decides the shape of the test. A two-sided test has H1: μ ≠ μ0 and rejects when the test statistic is far out in either tail. The significance level α is then split into α ÷ 2 in each tail. A one-sided test has H1: μ > μ0 or μ < μ0 and puts all of α in one tail. At 5 percent with a normal statistic the critical values are 1.96 for two-sided and 1.645 for one-sided. Use one-sided only when theory or the decision problem rules out the other direction before the data are seen, for example a risk limit that matters only when exceeded. Choosing the tail after seeing the sign of the estimate is a classic error.

A practice question

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

Which situation calls for a two-sided test?

  1. Testing whether a VaR model's exception rate exceeds its nominal rate, when only too many exceptions are a concern.Why not: Describes a directional concern in one tail, which calls for a one-sided test.
  2. Any test carried out at a 5 percent significance level.Why not: Confuses the significance level with the number of tails; the tails are set by the alternative hypothesis.
  3. Testing whether a hedge ratio differs from 1.00, where a deviation in either direction would matter. - correct
  4. Any test based on a large sample.Why not: Sample size affects the standard error, not whether the alternative is directional.

A two-sided test has an alternative of 'not equal to' and splits alpha between both tails. It fits a question in which departures above and below the hypothesized value both matter.

A harder one

A test of a mean has a test statistic z = 1.80, and the direction of the alternative (mu above the hypothesized value) was fixed before the data were collected. Which statement about a 5 percent test is correct?

  1. Both the one-sided and the two-sided tests reject, because 1.80 exceeds 1.645.Why not: Applies the one-sided critical value to the two-sided test, which requires 1.96.
  2. The one-sided test rejects (1.80 exceeds 1.645), but a two-sided test would not (1.80 is below 1.96). - correct
  3. Neither test rejects, because 1.80 is below the 1.96 cut-off.Why not: Applies the two-sided critical value to the one-sided test, which uses 1.645.
  4. The two-sided test rejects but the one-sided test does not, because two tails give two chances to reject.Why not: Reverses the effect; splitting alpha across two tails pushes the critical value out, making rejection harder.

At 5 percent the one-sided critical value is 1.645 and the two-sided value is 1.96. A statistic of 1.80 lies between them, so only the one-sided test rejects.

The hardest tier

A risk team pre-specified H0: mean P&L >= 0 versus H1: mean P&L < 0 for a desk. The sample gives a test statistic z = +1.55. The p-value of this one-sided test is closest to:

  1. 0.939 - correct
  2. 0.061Why not: Takes the upper-tail area although H1 points to the lower tail.
  3. 0.121Why not: Doubles the upper-tail area as for a two-sided test, and also uses the wrong tail.
  4. 0.879Why not: Computes one minus the two-sided p-value, which is not the one-sided lower-tail probability.

H1 is in the lower tail, so p = P(Z <= 1.55) = 0.939. The positive sample result is entirely consistent with H0 and gives no evidence of losses.

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