Answer to Question #217702 in Statistics and Probability for doreen

Question

A survey on economic status of seafarer showed that 70% are successful while 30% failed. A case history of 20 seafarer  are now under study. What is the probability that more than 12 of them are successful? Find the mean, variance and standard deviation.

Problem. Find the probability value of P(Z< - 0.5)

Answer 1

Let $X=$the number of successful seafarer: $X\sim Bin(n, p)$

Given $n=20, p=0.7, q=0.3.$

$$P(X=x)=\dbinom{20}{x}(0.7)^x(0.3)^{20-x}$$

$$P(X>12)=P(X=13)+P(X=14)$$

$$+P(X=15)+P(X=16)+P(X=17)$$

$$+P(X=18)+P(X=19)+P(X=20)$$

$$=0.77227179742$$

$$\mu=np=20(0.7)=14$$

$$Var(X)=\sigma^2=npq=20(0.7)(0.3)=4.2$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{4.2}\approx2.0494$$

$P(Z<-0.5)=0.308538$