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Example The height of boys at the age of 20 nationwide follows a normal distribution of N(171.9, 5.6^{2}). Extract 120 20-year-old boys and find the probability that the average height depends on 1.3cm or more from 171.9cm. Answer Let the average height of 120 boys be X̄. Then by the central limit theorem，

X̄ ∼ N(171.9, \frac{5.6^{2}}{120} )

Note that the probability that the average hight is 1.3cm taller than 171.9cm is given by

P_{r}(|X̄ − 171.9| ≥ 1.3) = 1 − 2P_{r}(0 ≤ X̄ − 171.9 < 1.3)

Here,

P_{r}(0 ≤ X̄−171.9 < 1.3) = P_{r}(0 ≤ \frac{X̄ − 171.9}{5.6/\sqrt{120}} ≤ \frac{1.3}{5.6/\sqrt{120}}) = P_{r}(0 ≤ Z ≤ 2.543) = 0.4945

implies that

P_{r}(|X̄ − 171.9| ≥ 1.3) = 1 − 2(0.4945) = 0.011