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

A sign test is applied to 12 paired differences (after-before) in monthly premium collections for a branch. Of the 12, none is zero; 9 are positive and 3 are negative. Under H0 (median difference = 0), the number of positive differences is Binomial(12, 0.5). Which expression gives the exact two-sided p-value?

The two-sided p-value is twice the upper tail probability P(X ≥ 9) for a Binomial(12, 0.5) variable. The binomial is symmetric under the null, so the opposite tail has equal probability, and the tail rather than a single point probability measures extremeness.

  1. A2 × P(X ≥ 9), where X ~ Bin(12, 0.5)Correct
  2. BP(X ≥ 9), where X ~ Bin(12, 0.5)
  3. C2 × P(X ≥ 9), where X ~ Bin(11, 0.5)
  4. D2 × P(X = 9), where X ~ Bin(12, 0.5)
  5. P(X ≥ 9) + P(X ≤ 9), where X ~ Bin(12, 0.5)

Explanation

The statistic is the number of positive signs, X ~ Bin(12, 0.5) under H0. The observed value 9 is in the upper tail, so the one-sided p-value is P(X ≥ 9). By symmetry of Bin(12, 0.5) the two-sided p-value doubles it. Using only one tail gives a one-sided p-value; using a point probability ignores the tail.

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