The National Vaccine Information Center estimates that 90% of Americans have had

Question

The National Vaccine Information Center estimates that 90% of Americans have had chickenpox by the time they reach adulthood. (a) Suppose we take a random sample of 100 American adults. Is the use of the binomial distribution appropriate for calculating the probability that exactly 97 had chickenpox before they reached adulthood? Explain. (b) Calculate the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood. (c) What is the probability that exactly 3 out of a new sample of 100 American adults have not had chickenpox in their childhood? (d) What is the probability that at least 1 out of 10 randomly sampled American adults have had chickenpox? The National Vaccine Information Center estimates that 90% of Americans have had chickenpox by the time they reach adulthood. (a) Suppose we take a random sample of 100 American adults. Is the use of the binomial distribution appropriate for calculating the probability that exactly 97 had chickenpox before they reached adulthood? Explain. (b) Calculate the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood. (c) What is the probability that exactly 3 out of a new sample of 100 American adults have not had chickenpox in their childhood? (d) What is the probability that at least 1 out of 10 randomly sampled American adults have had chickenpox? The National Vaccine Information Center estimates that 90% of Americans have had chickenpox by the time they reach adulthood. (a) Suppose we take a random sample of 100 American adults. Is the use of the binomial distribution appropriate for calculating the probability that exactly 97 had chickenpox before they reached adulthood? Explain. (b) Calculate the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood. (c) What is the probability that exactly 3 out of a new sample of 100 American adults have not had chickenpox in their childhood? (d) What is the probability that at least 1 out of 10 randomly sampled American adults have had chickenpox?

Explanation / Answer

a. Here we can see that all the four conditions of binomial distribution matches

1. fixed number of trials=100, fixed number of success =97, fixed probability of success=0.90 and also the independent trials.

So yes use of binomial distribution is appropriate for calculating probability that exactly 97 had chickenpox before they reached adulthood

b. P(x=97)=ncxp^x(1-p)^n-x=100c97*0.90&97*(0.10)^3=0.0059

c. For this p=0.10 so P(x=3)=100c3*0.10^3*0.90^97=0.0059

d. Here again p=0.10 and n=10

P(x>=1)=P(x=1)+P(x=2)....+P(x=10)

So P(x>=1)=0.6513

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