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Use confidence intervals in one-sided and two-sided tests, and state what a test's result means at a chosen confidence level.

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

The idea

Confidence intervals can carry out one-sided tests as well as two-sided ones. A two-sided test at significance α corresponds to a (1 − α) interval. A one-sided test at α puts the whole α in one tail, so it corresponds to a two-sided interval with this confidence level: Confidence = 1 − 2 × α. A 5 percent one-sided test therefore uses the 90 percent two-sided interval, and a 1 percent one-sided test uses the 98 percent interval. For H1: μ > μ0, reject if the lower limit exceeds μ0. For H1: μ < μ0, reject if the upper limit is below μ0. Rejecting at the 95 percent confidence level means significance of 5 percent. Using the 95 percent interval for a one-sided 5 percent test is the common slip, and it makes you fail to reject too often.

A practice question

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

Which two-sided confidence interval corresponds to a one-sided hypothesis test at the 5 percent significance level?

  1. A 95 percent confidence intervalWhy not: Matches the interval to one minus alpha without noting that a two-sided interval splits alpha across two tails.
  2. A 90 percent confidence interval - correct
  3. A 97.5 percent confidence intervalWhy not: Uses 1 - alpha/2, which is the level appropriate for a two-sided test at 5 percent.
  4. A 99 percent confidence intervalWhy not: Uses the complement of a 1 percent level, unrelated to a 5 percent one-sided test.

A one-sided 5 percent test leaves 5 percent in one tail. A 90 percent two-sided interval also leaves 5 percent in each tail, so its relevant limit matches the one-sided test.

A harder one

A sample mean excess return is 0.74 percent with a standard error of 0.40 percent. At the 5 percent level, test H0: mu <= 0 against H1: mu > 0 using a confidence interval. Which conclusion is correct?

  1. Fail to reject: the 95 percent two-sided interval is about -0.04 to 1.52 percent and contains zero.Why not: Uses the 95 percent interval for a one-sided 5 percent test; the matching interval is 90 percent.
  2. Fail to reject: the 99 percent two-sided interval is about -0.29 to 1.77 percent and contains zero.Why not: Uses the 99 percent interval, which corresponds to a 0.5 percent one-sided test, not a 5 percent one.
  3. No decision is possible, because an interval can only be used for two-sided tests.Why not: Overlooks that the one-sided test uses a one-sided limit of the interval at the matching confidence level.
  4. Reject: the lower limit of the 90 percent two-sided interval is about 0.08 percent, above zero. - correct

A one-sided 5 percent test matches the 90 percent interval: 0.74 +/- 1.645 x 0.40 = (0.08, 1.40). The lower limit exceeds zero, so H0: mu <= 0 is rejected.

The hardest tier

An analyst's 95 percent two-sided interval for a mean return (estimate 1.1 percent, standard error 0.6 percent) is about -0.08 to 2.28 percent. She says that a 5 percent one-sided test of H0: mu <= 0 against H1: mu > 0 must also fail to reject because the interval contains zero. Evaluate her claim.

  1. She is right: the 95 percent interval is the relevant interval for any test at the 5 percent level.Why not: Ignores that a one-sided 5 percent test places all alpha in one tail and so uses the narrower 90 percent interval.
  2. She is right, because a one-sided test is stricter than a two-sided test and rejects less often.Why not: Reverses the relationship; the one-sided critical value is smaller, so rejection in the stated direction is easier.
  3. She is right: the one-sided p-value is 0.067, which exceeds 0.05.Why not: Uses the two-sided p-value of 0.067 for a one-sided test; the one-sided p-value is 0.033.
  4. She is wrong: the matching interval is the 90 percent one, about 0.11 to 2.09 percent, which excludes zero. - correct

z = 1.1/0.6 = 1.83, which exceeds 1.645 but not 1.96. The one-sided p-value is 0.033, below 0.05, so the one-sided test rejects even though the two-sided 95 percent interval includes zero.

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