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

A sample of 10 claim amounts from a normal population has sample variance 18. To test H0: sigma^2 = 12 against H1: sigma^2 > 12, which statistic and distribution under H0 should be used, and what is its value?

The statistic is 13.5 on 9 degrees of freedom. It is (n-1) times the sample variance divided by the hypothesised variance, 9 times 18 over 12, and follows a chi-squared distribution with n-1 degrees of freedom under the null hypothesis.

  1. A13.5 on 9 degrees of freedomCorrect
  2. B13.5 on 10 degrees of freedom
  3. C15.0 on 9 degrees of freedom
  4. D6.67 on 9 degrees of freedom
  5. 1.50 on 9 degrees of freedom

Explanation

The statistic is (n-1)s^2/sigma0^2 = 9*18/12 = 13.5, which follows chi-squared with 9 degrees of freedom under H0. Using n instead of n-1 gives 15 and the wrong degrees of freedom gives option B. 1.50 is just the variance ratio, and 6.67 inverts the ratio incorrectly.

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