IAI Actuarial Core Principles · Actuarial Statistics · Random sampling and sampling distributions
For a random sample of size n from any population with finite variance σ², which statement about the sample variance S² with divisor n − 1 is correct?
E(S²) = σ² for any population with finite variance. The sum of squared deviations about the sample mean has expectation (n−1)σ², so dividing by n−1 removes the bias, and normality is not required.
- AE(S²) = σ² (n−1)/n
- BE(S²) = σ²Correct
- CE(S²) = σ² n/(n−1)
- DE(S²) = σ²/n
- E(S²) = σ² only if the population is normal
Explanation
Since E[Σ(Xi−X̄)²] = (n−1)σ² for any population with finite variance, dividing by n−1 gives an unbiased estimator. The factor (n−1)/n applies to the divisor-n version. Normality is not needed for unbiasedness.
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