next up previous
Next: この文書について...

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 $\bar{X}$. Then by the central limit theorem，

$\displaystyle \bar{X} \sim N(171.9, \frac{5.6^{2}}{120})$

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

$\displaystyle P_{r}(\vert\bar{X} - 171.9\vert \geq 1.3) = 1 - 2P_{r}(0 \leq \bar{X} - 171.9 < 1.3)$

Here,

$\displaystyle P_{r}(0 \leq \bar{X} - 171.9 < 1.3) = P_{r}(0 \leq \frac{\bar{X} ... ...rt{120}} \leq \frac{1.3}{5.6/\sqrt{120}}) = P_{r}(0 \leq Z \leq 2.543) = 0.4945$

implies that

$\displaystyle P_{r}(\vert\bar{X} - 171.9\vert \geq 1.3) = 1 - 2(0.4945) = 0.011$