표본평균의 확률

문제의 조건에 의해 확률변수 $\displaystyle \overline{X}$는 정규분포 $\displaystyle {\mathrm{N}} \left( 50 , \frac{2 ^{2}}{9} \right)$, 즉 $\displaystyle {\mathrm{N}} \left( 50 , \left( \frac{2}{3} \right) ^{2} \right)$을 따르므로 $\displaystyle \mathrm{P} \mathit{(} 49 \leq \overline{X} \leq 51 )$$\displaystyle = \mathrm{P} \mathit{\left( \frac{49 - 50}{\frac{2}{3}} \leq Z \leq \frac{51 - 50}{\frac{2}{3}} \right)}$ $\displaystyle = \mathrm{P} \mathit{\left( - \frac{3}{2} \leq Z \leq \frac{3}{2} \right)}$$= \mathrm{P} \mathit{(} 0 \leq Z \leq 1.5 )$ $= 2 \times 0.4332 = 0.8664$