Respuesta :
Answer:
0.1745
Step-by-step explanation:
We are given the following information:
We treat alumni contacted by telephone making a contribution of at least $50 as a success.
P(Alumni making a contribution) = 80% = 0.8
Then the number of alumni follows a binomial distribution, where
[tex]P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}[/tex]
where n is the total number of observations, x is the number of success, p is the probability of success.
Sample size, n = 20
We have to evaluate:
[tex]P(x = 15)\\= \binom{20}{15}(0.8)^{15}(1-0.8)^5\\=0.1745[/tex]
Thus, 0.1745 is the probability that exactly 15 alumni will make a contribution of at least $50 when contacted by telephone.
Using the binomial distribution, it is found that there is a 0.1746 = 17.46% probability that exactly 15 alumni will make a contribution of at least $50.
What is the binomial distribution formula?
The formula is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- n is the number of trials.
- p is the probability of a success on a single trial.
In this problem:
- 80% of the alumni contacted in this manner will make a contribution of at least $50, hence p = 0.8.
- A random sample of 20 alumni is selected, hence n = 20.
The probability that exactly 15 alumni will make a contribution of at least $50 is P(X = 15), hence:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 15) = C_{20,15}.(0.8)^{15}.(0.2)^{5} = 0.1746[/tex]
0.1746 = 17.46% probability that exactly 15 alumni will make a contribution of at least $50.
More can be learned about the binomial distribution at https://brainly.com/question/24863377
