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

A random sample of size 25 is taken from a population with variance 100. Using the sample variance S^2 with divisor (n-1), which of the following is the value of E[S^2] and the variance of the sample mean respectively?

E[S^2] is 100 and the variance of the sample mean is 4. The sample variance with divisor n-1 is unbiased for the population variance, and the variance of the mean is sigma^2/n = 100/25 = 4. Using divisor n would give 96 instead.

  1. A100 and 4Correct
  2. B96 and 4
  3. C100 and 20
  4. D100 and 0.16
  5. 96 and 100

Explanation

S^2 with divisor n-1 is unbiased, so E[S^2] = 100. Var(mean) = 100/25 = 4. The value 96 arises from wrongly using the divisor n, which gives (n-1)/n x 100 = 96, and that applies only to the biased version.

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