Answer to Question #136812 in Statistics and Probability for Princess

Question

Let X~Bin(50, 0.64). For samples of size 36, calculate P(31.28 < X bar < 33.36).

Answer 1

$X\sim Bin(50, 0.64)$

$$\mu=np=50\cdot0.64=32$$

$$\sigma=\sqrt{np(1-p)}=\sqrt{50(0.64)(1-0.64)}=2.4\sqrt{2}$$

$\bar{X}\sim N(32, (2.4\sqrt{2})^2)$

$n_1=36$

$$P(31.28<\bar{X}<33.36)=P(\bar{X}<33.36)-P(\bar{X}<31.28)=$$

$$=P(Z<\dfrac{33.36-32}{(2.4\sqrt{2})/\sqrt{36}})-P(Z<\dfrac{31.28-32}{(2.4\sqrt{2})/\sqrt{36}})\approx$$

$$\approx P(Z<2.404163)-P(Z<-1.272792)\approx$$

$$\approx0.991895-0.101546=0.890349$$

$P(31.28<\bar{X}<33.36)=0.890349$