Respuesta :
Answer:
The answer to this question would normally be, 85.96%, though that isnt an answer choice so the closest thing to it would most likely be answer choice A.
Using the binomial approximation to the normal, it is found that:
0.86 probability that at most 60 say yes, given by option a.
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Binomial probability distribution
Probability of exactly x successes on n repeated trials, with p probability.
The expected value is:
[tex]E(X) = np[/tex]
The standard deviation is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
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Normal probability distribution
Problems of normally distributed distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The Z-score measures how many standard deviations the measure is from the mean.
- Each z-score has a corresponding p-value.
- The p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.
- The probability that the measure is greater than X is 1 subtracted by the p-value.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex], if [tex]np \geq 10[/tex] and [tex]n(1-p) \geq 10[/tex].
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- 64% use services, thus, [tex]p = 0.64[/tex]
- 87 people are sampled, thus, [tex]n = 87[/tex]
The mean and standard deviation are given by:
[tex]\mu = E(X) = np = 87(0.64) = 55.68[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{87(0.64)(0.36)} = 4.4771[/tex]
Using continuity correction, the probability that at most 60 use is [tex]P(X \leq 60 + 0.5) = P(X \leq 60.5)[/tex], which is the p-value of Z when X = 60.5, thus:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{60.5 - 55.68}{4.4771}[/tex]
[tex]Z = 1.08[/tex]
[tex]Z = 1.08[/tex] has a p-value of 0.86.
Thus, 0.86 probability that at most 60 say yes, given by option a.
A similar problem is given at https://brainly.com/question/16178115
