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

A random sample X1,...,X25 is drawn from a population with mean 80 and variance 100. The sample mean is denoted X-bar. What are the mean and standard deviation of X-bar?

The sample mean has mean 80 and standard deviation 2. Its variance is the population variance 100 divided by n=25, giving 4, and the square root of 4 is 2. The expected value equals the population mean.

  1. AMean 80, standard deviation 10
  2. BMean 80, standard deviation 2Correct
  3. CMean 16, standard deviation 2
  4. DMean 80, standard deviation 4
  5. Mean 3.2, standard deviation 0.4

Explanation

E(X-bar)=80. Var(X-bar)=100/25=4, so the standard deviation is 2. Option with 10 forgets to divide by root n; option with 4 reports the variance instead of the standard deviation.

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