Respuesta :
Answer:
What is the probability that among 20 randomly selected US adult workers 14 have a high school diploma?
b) 0.1712
How many US adult workers are expected to have a high school diploma?
c) 13 US adult workers
What is the standard deviation of the number of US adult workers with high school diploma?
a) 2.13 US adult workers
Step-by-step explanation:
For each adult worker, there are only two possible outcomes. Either they have a high school diploma, or they do not. The probability of an adult worker having a high school diploma is independent of other adult workers. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
According to the US Department of Labor, 65% of all adult workers have a high school diploma
This means that [tex]p = 0.65[/tex]
What is the probability that among 20 randomly selected US adult workers 14 have a high school diploma?
This is P(X = 14). So
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 14) = C_{20,14}.(0.65)^{14}.(0.35)^{6} = 0.1712[/tex]
How many US adult workers are expected to have a high school diploma?
[tex]E(X) = np = 20*0.65 = 13[/tex]
What is the standard deviation of the number of US adult workers with high school diploma?
[tex]\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{20*0.65*0.35} = 2.13[/tex]
