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

A random sample of size 9 is taken from a normal population with unknown mean and variance σ² = 16. Which is the distribution of 8S²/16, where S² is the unbiased sample variance?

The quantity 8S²/16 follows a chi-squared distribution with 8 degrees of freedom. For normal samples, (n−1)S²/σ² is chi-squared with n−1 degrees of freedom, and here n−1 = 8.

  1. AChi-squared with 8 degrees of freedomCorrect
  2. BChi-squared with 9 degrees of freedom
  3. Ct with 8 degrees of freedom
  4. DNormal with mean 8 and variance 16
  5. F with 8 and 9 degrees of freedom

Explanation

For a normal sample, (n−1)S²/σ² follows a chi-squared distribution with n−1 degrees of freedom. Here n−1 = 8 and σ² = 16, so 8S²/16 is chi-squared with 8 degrees of freedom. Using 9 would apply if the true mean were known and used.

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