Respuesta :
Answer:
The mean number favoring the substation is 12 citizens.
Step-by-step explanation:
For each citizen surveyed, there are only two possible outcomes. EIther they favored building a police substation in their neighborhood, or they opposed. This means that we use the binomial probability distribution to solve this problem.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, with p probability, and X can only have two outcomes.
Has an expected value of:
[tex]E(X) = np[/tex]
In this problem, we have that:
[tex]n = 15, p = 0.8[/tex]
E(X) = 15*0.8 = 12.
The mean number favoring the substation is 12 citizens.
Mean of a binomial distribution is sample size times probability of success. The mean number favoring the substation is 12
How to find that a given condition can be modeled by binomial distribution?
Binomial distributions consists of n independent Bernoulli trials.
Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))
Suppose we have random variable X pertaining binomial distribution with parameters n and p, then it is written as
[tex]X \sim B(n,p)[/tex]
The probability that out of n trials, there'd be x successes is given by
[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}[/tex]
Also,
E(X) = mean value of X = np
For the given case, it is known that a person can be either favoring the substation or not. And that probability of favoring the substation is 80% = 0.8
Thus, taking the random variable X tracking the number of people who support substation, we get X as to be pertaining binomial distribution with n = 15, and p = probability of success = 0.8
(where success is when person supports substation, and since all people's decision are assumed to be independent, they form independent Bernoulli trials).
Thus,
[tex]X \sim B(n,p)\\X \sim B(15, 0.8)[/tex]
The mean number of people favoring the substation is mean of X which is E(X)
We get [tex]E(X) = np = 15 \times 0.8 = 12[/tex]
Thus, the mean number favoring the substation is 12
Learn more about binomial distribution here:
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