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

X1,...,X10 are independent N(mu, sigma^2) observations, and S^2 is the sample variance with divisor n-1. Which statement about the distribution of 9S^2/sigma^2 is correct?

It follows a chi-squared distribution with 9 degrees of freedom. For a normal sample of size 10, (n-1)S^2/sigma^2 is chi-squared with n-1 degrees of freedom, because the sample mean is estimated from the data.

  1. AChi-squared with 10 degrees of freedom
  2. Bt with 9 degrees of freedom
  3. CChi-squared with 9 degrees of freedomCorrect
  4. DNormal with mean 9 and variance 18
  5. F with 9 and 10 degrees of freedom

Explanation

For a normal sample, (n-1)S^2/sigma^2 follows a chi-squared distribution with n-1 = 9 degrees of freedom. One degree of freedom is lost because the sample mean is estimated. Chi-squared with 10 df applies only when the true mean is known and used in the sum of squares.

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