Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

More than a decade ago, high levels of lead in the blood put 84% of children at risk. A concerted effort was made to remove lead from the environment. Now, suppose only 8% of children in the United States are at risk of high blood-lead levels.

(a) In a random sample of 212 children taken more than a decade ago, what is the probability that 50 or more had high blood-lead levels? (Round your answer to three decimal places.)


(b) In a random sample of 212 children taken now, what is the probability that 50 or more have high blood-lead levels? (Round your answer to three decimal places.)

Answer 1

a)

n=

212

p=

0.8400

here mean of distribution=μ=np=		178.08
and standard deviation σ=sqrt(np(1-p))=		5.34
for normal distribution z score =(X-μ)/σx

therefore from normal approximation of binomial distribution and continuity correction:

probability =P(X>49.5)=P(Z>(49.5-178.08)/5.338)=P(Z>-24.09)=1-P(Z<-24.09)=1-0=1.000

b)

for p=0.08:

here mean of distribution=μ=np=		16.96
and standard deviation σ=sqrt(np(1-p))=		3.95

probability =P(X>49.5)=P(Z>(49.5-16.96)/3.95)=P(Z>8.24)=1-P(Z<8.24)=1-1=0.000