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

A graduation of mortality rates is tested over 20 ages. The chi-squared test is passed comfortably, but 17 of the 20 deviations (actual minus expected deaths) are positive. Which statement is most accurate?

The signs test would indicate bias, with graduated rates systematically too low. Seventeen positive deviations out of twenty is very unlikely under a fair-coin binomial, even though the chi-squared test can still be passed because it ignores the direction of the deviations.

  1. AThe graduation is satisfactory because the chi-squared test was passed
  2. BThe signs test would indicate the graduated rates are biased, systematically too lowCorrect
  3. CThe signs test would indicate the graduated rates are too high
  4. DThe runs test must be failed because the signs are unequal
  5. The serial correlations test would show the deviations are independent

Explanation

Positive deviations mean actual deaths exceed expected deaths, so graduated rates are too low. With 17 of 20 positive, under a Binomial(20, 0.5) the result is very unlikely, so the signs test would reject. The chi-squared test can pass despite such bias, and the unequal count alone does not determine the runs test.

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