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

Over 12 age groups, a graduation is tested using the chi-squared test. The graduated rates were obtained by fitting a formula with 3 estimated parameters, and the observed statistic is 14.0. What are the degrees of freedom to use and the conclusion at the 5% level? (Critical values at 5%: chi-squared with 9 df is 16.92, with 12 df is 21.03, with 11 df is 19.68, with 3 df is 7.81.)

Use 9 degrees of freedom, being 12 groups less 3 fitted parameters. The critical value is 16.92, and since the statistic of 14.0 is lower, the graduation is not rejected at the 5% level.

  1. A12 df; do not reject, since 14.0 is below 21.03
  2. B9 df; do not reject, since 14.0 is below 16.92Correct
  3. C9 df; reject, since 14.0 exceeds 7.81
  4. D3 df; reject, since 14.0 exceeds 7.81
  5. 11 df; do not reject, since 14.0 is below 19.68

Explanation

When parameters are fitted to the data, degrees of freedom are the number of age groups minus the number of fitted parameters: 12 - 3 = 9. The 5% critical value is 16.92 and 14.0 is below it, so the graduation is not rejected. Using 12 df ignores the fitted parameters.

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