Question

According to a​ survey, 2323​% of residents of a country 25 years old or older had earned at least a​ bachelor's degree. You are performing a study and would like at least 1010 people in the study to have earned at least a​ bachelor's degree. ​(a) How many residents of the country 25 years old or older do you expect to randomly​ select? ​(b) How many residents of the country 25 years old or older do you have to randomly select to have a probability 0.9450.945 that the sample contains at least 1010 who have earned at least a​ bachelor's degree? ​(a) The number of randomly selected residents is 4444. ​(Round up to the nearest​ integer.) ​(b) The number of randomly selected​ residents, with a probability 0.9450.945 containing at least 1010 who have earned at least a​ bachelor's degree, is 2828. ​(Round up to the nearest​ integer.)

Answer 1

a) 44

b) 64

Explanation:

Applying the central limit theorem, the proportion of the random sample of 25 years old or older that have earned at least a​ bachelor's degree will be equal to the population proportion of 23%.

Mean = np

Mean = 10

p = 0.23

n = ?

10 = n×0.23

N

n = (10/0.23) = 43.5 = 44 to the nearest whole number.

b) This is a binomial distribution problem with the probability known and the number of trials unknown.

Binomial distribution function is represented by

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = ?

x = Number of successes required = 10

p = probability of success = 0.23

q = probability of failure = 1 - 0.23 = 0.77

P(X ≥ 10) = 0.945

P(X ≥ 10) = 1 - P(X < 10) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9)]

0.945 = 1 - [Σ P(X=x)] (with the summation of x from 0 to 9)

Using the trial and error method on the binomial distribution formula calculator, n is obtained to be 64.

Hope this Helps!!!