Question

According to the American Community Survey, 27% of residents of the United States 25 years old or older had earned a bachelor degree. Suppose you select 12 residents of the United States 25 years old or older and recorded the number who had earned a bachelor degree. Round probabilities to 4 decimal places.

1. Explain why this is a binomial experiment.
2. Find and interpret the probability that exactly 5 of them had a bachelor degree.
3. Find and interpret the probability that fewer than 5 of them had a bachelor degree.
4. Find and interpret the probability that at least 5 of them had a bachelor degree.
5. Compute the mean and standard deviation of the binomial random variable.

Answer 1

(A) It is a binomial experiment because it has independent sample, fixed number (n = 12) with p= 0.27

there are only two possible outcomes for each event and each event is random with same probability of success

So, it is a binomial experiment

(B) Using binompdf

setting n = 12, p = 0.27 and k = 5

P(x=5) =binompdf(n,p,k)

= binompdf(12,0.27,5)

= 0.1255

(C)

Using binompdf

setting n = 12, p = 0.27 and k = 5

P(x<5) =binompdf(n,p,k-1)

= binompdf(12,0.27,5-1)

= 0.7984

(D) Probability of at least 5 = 1 - (probability of fewer than 5)

= 1 - 0.7984 (using part (c) answer)

= 0.2016

(E) Mean = n*p

setting n = 12 and p = 0.27

E[x] = mean = 12*0.27 = 3.24

and

standard deviation = sqrt{n*p*q}

setting n = 12, p = 0.27 and q = 1- 0.27 = 0.73

we get

standard deviation = sqrt(12*0.27*0.73)

= 1.54