Question

The National Vaccine Information Center estimates that 91.2% of Americans have had chickenpox by the time they reach adulthood.

(a) Calculate the probability that exactly 97 out of 100 randomly sampled American adults had chickenpox during childhood.

(b) What is the probability that exactly 3 out of a new sample of 100 American adults have not had chickenpox in their childhood?

(c) What is the probability that at least 1 out of 10 randomly sampled American adults have had chickenpox?

(d) What is the probability that at most 3 out of 10 randomly sampled American adults have not had chickenpox? We now consider a random sample of 120 American adults.

(e) How many people in this sample would you expect to have had chickenpox in their childhood? And with what standard deviation?

(f) What is the probability that 96 or fewer people in this sample have had chickenpox in their childhood? (round to 4 places)

Answer 1

The sample size of n American adults will have a binomial distribution with parameters n and p. The probability mass function is defined as,

(a)

Sample size, n = 100

Probability (have had chickenpox) = 0.912

Number of success, k = 97

For calculation purposes, the probability is obtained in excel using the function =BINOM.DIST(). The screenshot is shown below,

(b)

Sample size, n = 100

Probability (have not had chickenpox) = 1 - 0.912 = 0.088

Number of success, k = 3

the probability is obtained in excel using the function =BINOM.DIST(). The screenshot is shown below,

(c)

Sample size, n = 10

Probability (have had chickenpox) = 0.912

Number of success, k >= 1

the probability is obtained in excel using the function =BINOM.DIST(). The screenshot is shown below,

(d)

Sample size, n = 120

Probability (have not had chickenpox), p = 0.088

Number of success, k <= 120*3/10= 36

the probability is obtained in excel using the function =BINOM.DIST(). The screenshot is shown below,

(e)

n = 120

p = 0.912

(f)

Sample size, n = 120

Probability (have had chickenpox), p = 0.912

Number of success, k <= 96