Historically about 51 of the babies born in the United State

Historically about 51% of the babies born in the United States are boys. That is, we know that p=0.51. A survey plans to study 650 randomly selected newborns.

What are the mean and standard deviation of the statistic p hat , the sample proportion who are boys?

Use the Normal approximation to find the probability that p hat would be less than 0.5.

Solution

n=650

p = 0.51

mean = n*p = 0.51 * 650 = 331.5

sd = sqrt(n*p*q) = 12.74

x=0.5 * 650 = 325

z=(x-m)/sd

P(x<325)=P(z<(325-331.5)/12.74)

P(x<325) = P(z<-0.51)

P(x<325) = 0.3050