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IAI Actuarial Core Principles · Actuarial Statistics · Central limit theorem

A sample mean of n independent observations from a distribution with finite variance is approximately normal for large n. Which statement about the central limit theorem is correct?

The correct statement is that the standardised sample mean converges in distribution to a standard normal as n grows. This holds for independent identically distributed variables with finite variance, whatever their original shape; the individual observations themselves do not become normal.

  1. AIt requires the underlying distribution to be normal
  2. BThe standardised sample mean converges in distribution to a standard normal as n increasesCorrect
  3. CThe sample variance of the data shrinks to zero as n increases
  4. DThe distribution of individual observations becomes normal as n increases
  5. It applies only when the observations are Poisson

Explanation

The CLT states that (Xbar - mu)/(sigma/sqrt n) tends in distribution to N(0,1) for independent identically distributed variables with finite variance. The individual observations do not change shape, and the underlying distribution need not be normal.

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