Respuesta :
Answer:
Step-by-step explanation:
a) About 25% of those called will find an excuse (work, poor health, travel out of town, etc. to avoid jury duty. So, the probability that a jury will be available to serve is ; P = 1 - q = 1 - 0.25 = 0.75
Given n = 15 and p = 0.75
P(X = 15) = Binompdf(15, 0.75, 15) = 0.013
b) Event "9 or more will not be available to serve on the jury" is equivalent to "6 or less will be available to serve".
P(X ≤ 6) = Binompdf(15, 0.75, 6) = 0.004
c) The expected number and the standard deviation of those available to serve on the jury are: μ = np = 15(0.75) = 11.25
σ = √np(1 - p) = √15(0.75)(1 - 0.75) = 1.68
d) Use function:
P(X ≥ 12) = 1 - P(X ≤ 11) = 1 - Binompdf(n, 0.75, 11)
By using above function, the probabilities of P(X ≥ 12) at different sample size are shown below:
n P(X ≥ 12)
12 0.032
13 0.127
14 0.281
15 0.461
16 0.630
17 0.765
18 0.861
19 0.923
20 0.959
21 0.979
Therefore, The jury commissioner must contact at least 20 people to be 95.9% sure of finding at least 12 people who are available to serve.
Using the binomial distribution, it is found that:
a) There is a 0.013 = 1.3% probability that all 15 will be available to serve on the jury.
b) There is a 0.943 = 94.3% probability that 9 or more will not be available to serve on the jury.
c) The expected number of those available to serve on the jury is 11.25, with a standard deviation of 1.68.
d) The jury commissioner must select 20 people.
For each person, there are only two possible outcomes, either they are available to serve on the jury, or they are not. The probability of a person being available to serve on the jury is independent of any other person, hence, the binomial distribution is used to solve this question.
Binomial probability distribution
[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:
- 25% of those called will find an excuse, hence 75% will be available, hence [tex]p = 0.75[/tex]
- 15 people are called for jury duty, hence [tex]n = 15[/tex].
Item a:
This probability is P(X = 15), hence:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 15) = C_{15,15}.(0.75)^{15}.(0.25)^{0} = 0.013[/tex]
There is a 0.013 = 1.3% probability that all 15 will be available to serve on the jury.
Item b:
This probability is:
[tex]P(X \geq 9) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)[/tex]
Using the same formula as in item a to find each of these probabilities, and then adding them, it is found that:
[tex]P(X \geq 9) = 0.943[/tex]
There is a 0.943 = 94.3% probability that 9 or more will not be available to serve on the jury.
Item c:
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation is:
[tex]\sqrt{V(X)} = \sqrt{np(1 - p)}[/tex]
Hence:
[tex]E(X) = np = 15(0.75) = 11.25[/tex]
[tex]\sqrt{V(X)} = \sqrt{np(1 - p)} = \sqrt{15(0.75)(0.25)} = 1.68[/tex]
The expected number of those available to serve on the jury is 11.25, with a standard deviation of 1.68.
Item d:
Using a binomial calculator, we keep increasing n by 1 until [tex]P(X \geq 12) = 0.959[/tex], hence, n = 20 is needed, that is:
- The jury commissioner must select 20 people.
To learn more about the binomial distribution, you can take a look at https://brainly.com/question/24863377
