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IAI Actuarial Core Principles · Actuarial Statistics · Random sampling and sampling distributions

A random sample of size 10 is drawn from a normal population with unknown mean and variance 4. The sample variance is S^2 = 6 (divisor n-1). Which of the following is the value of the statistic 9S^2/σ^2 and its distribution?

The statistic is 9×6/4 = 13.5, and it follows a chi-square distribution with 9 degrees of freedom, because (n-1)S^2/σ^2 has n-1 degrees of freedom when sampling from a normal population.

  1. A13.5, chi-square with 9 degrees of freedomCorrect
  2. B13.5, chi-square with 10 degrees of freedom
  3. C15, chi-square with 9 degrees of freedom
  4. D9.0, chi-square with 9 degrees of freedom
  5. 13.5, t with 9 degrees of freedom

Explanation

(n-1)S^2/σ^2 follows chi-square with n-1 = 9 degrees of freedom. Value = 9×6/4 = 13.5. Using 10 degrees of freedom or dividing wrongly gives the other options; the statistic is not a t variable.

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