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FRM Part I · FRM Exam Part I · Common Univariate Random Variables

Which statement about the Central Limit Theorem as applied to the sample mean of i.i.d. observations with finite variance is correct?

The Central Limit Theorem says that for i.i.d. observations with finite variance, the standardized sample mean converges to a standard normal distribution as the sample size increases. It describes the sampling distribution of the mean, not the distribution of the individual data points.

  1. AThe distribution of individual observations becomes normal as the sample size grows
  2. BThe distribution of the standardized sample mean approaches a standard normal distribution as the sample size growsCorrect
  3. CThe sample mean is normally distributed for any sample size only if the population is skewed
  4. DThe standard error of the sample mean increases in proportion to the square root of n

Explanation

The CLT concerns the sampling distribution of the standardized sample mean, which converges to N(0,1) as n grows. It does not change the distribution of the individual observations. The standard error falls, not rises, with the square root of n.

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