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IAI Actuarial Core Principles · Actuarial Statistics · Hypothesis testing and goodness of fit

A permutation test compares mean claim settlement times of two groups, 3 observations in group X and 4 in group Y (7 in total, no ties). The test statistic is the difference in group means, and the observed arrangement gives the largest difference of all arrangements in the positive direction, with no other arrangement equalling it. What is the exact one-sided p-value, and the smallest achievable two-sided p-value assuming symmetry of the permutation distribution?

There are C(7,3) = 35 equally likely arrangements under the null hypothesis. A unique most extreme arrangement gives a one-sided p-value of 1/35, and doubling for the symmetric opposite tail gives a two-sided p-value of 2/35.

  1. A1/35 and 2/35Correct
  2. B1/7 and 2/7
  3. C1/35 and 1/35
  4. D1/21 and 2/21
  5. 1/70 and 1/35

Explanation

The number of ways to choose which 3 of 7 observations form group X is C(7,3) = 35, all equally likely under H0. If the observed arrangement is the unique most extreme in this direction, the one-sided p-value is 1/35. Symmetry gives an equally extreme arrangement in the other tail, so the two-sided p-value is 2/35. Using C(7,2)=21 or 70 miscounts the arrangements.

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