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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Graduation and graduation tests

A graduated mortality table covers 20 age groups. Under the null hypothesis that the graduated rates are correct, the standardised deviations are independent N(0,1). In a signs test, 15 of the 20 deviations are positive. Under the null the number of positive signs follows Binomial(20, 0.5), with P(X≥15)=0.0207. Using a two-sided test at the 5% level, what is the conclusion?

Reject the null hypothesis. The upper-tail probability of 0.0207 is doubled for a two-sided test, giving about 0.041, which is below the 5% significance level, so the excess of positive deviations suggests the graduation is biased.

  1. AReject, because the two-sided p-value is about 0.041, which is below 0.05Correct
  2. BDo not reject, because 0.0207 exceeds 0.025
  3. CReject, because 15 exceeds the expected count of 10 by more than one standard deviation of 2
  4. DDo not reject, because the two-sided p-value is 0.0207 and signs tests need a 1% level
  5. Do not reject, because the test is invalid with 20 groups

Explanation

For a two-sided test, double the tail probability: 2 × 0.0207 = 0.0414. This is below 0.05, so reject the null. Option 1 is wrong because 0.0207 is below 0.025 and the comparison is made against the doubled value.

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