표본비율 확률 비교

세 확률변수 $X$, $Y$, $W$는 각각 정규분포 $\mathrm{N} \left( 20 , 4 ^{2} \right)$, $\mathrm{N} \left( 45 , 6 ^{2} \right)$, $\mathrm{N} \left( 80 , 8 ^{2} \right)$을 따른다. 또, 확률변수 $Z$가 표준정규분포를 따른다고 하자. ㄱ. $\displaystyle \mathrm{P} \left( \left| \frac{X - 20}{100} \right| < \frac{1}{10} \right) =$$\displaystyle \mathrm{P} \left( \left| \frac{X - 20}{4} \right| < \frac{25}{10} \right) = \mathrm{P} \left( | Z | < 2.5 \right)$ $\displaystyle \mathrm{P} \left( \left| \frac{W - 80}{400} \right| < \frac{1}{10} \right) =$$\displaystyle \mathrm{P} \left( \left| \frac{W - 80}{8} \right| < \frac{50}{10} \right) = \mathrm{P} \left( | Z | < 5 \right)$ ∴ $\displaystyle \mathrm{P} \left( \left| \frac{X - 20}{100} \right| < \frac{1}{10} \right) <$$\displaystyle \mathrm{P} \left( \left| \frac{W - 80}{400} \right| < \frac{1}{10} \right)$ (참) ㄴ. $\displaystyle \mathrm{P} \left( \left| \frac{Y - 40}{225} \right| < \frac{1}{25} \right) =$$\displaystyle \mathrm{P} \left( \left| \frac{Y - 40}{6} \right| < \frac{225}{6 \cdot 25} \right) = \mathrm{P} \left( | Z | < 1.5 \right)$ ∴ $\displaystyle \mathrm{P} \left( \left| \frac{X - 20}{100} \right| < \frac{1}{10} \right) >$$\displaystyle \mathrm{P} \left( \left| \frac{Y - 40}{225} \right| < \frac{1}{25} \right)$ (거짓) ㄷ. $\displaystyle \mathrm{P} \left( \left| \frac{W - 80}{400} \right| < \frac{1}{25} \right) =$$\displaystyle \mathrm{P} \left( \left| \frac{W - 80}{8} \right| < \frac{50}{25} \right) = \mathrm{P} \left( | Z | < 2 \right)$ ∴ $\displaystyle \mathrm{P} \left( \left| \frac{Y - 40}{225} \right| < \frac{1}{25} \right)$$\displaystyle < \mathrm{P} \left( \left| \frac{W - 80}{8} \right| < \frac{50}{25} \right)$ (참) 따라서 옳은 것은 ㄱ, ㄷ이다.