Respuesta :
Answer:
The probability that exactly 27 of 104 eligible voters voted is 0.057 = 5.7%.
Step-by-step explanation:
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
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]
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 zscore 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. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
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].
In this case, assume that 104 eligible voters aged 18-24 are randomly selected.
This means that [tex]n = 104[/tex].
Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted.
This means that [tex]p = 0.22[/tex]
Mean and standard deviation:
[tex]\mu = 104*0.22 = 22.88[/tex]
[tex]\sigma = \sqrt{104*0.22*0.78} = 4.2245[/tex]
Probability that exactly 27 voted
By continuity continuity, 27 consists of values between 26.5 and 27.5, which means that this probability is the p-value of Z when X = 27.5 subtracted by the p-value of Z when X = 26.5.
X = 27.5
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{27.5 - 22.88}{4.2245}[/tex]
[tex]Z = 1.09[/tex]
[tex]Z = 1.09[/tex] has a p-value of 0.8621
X = 26.5
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{26.5 - 22.88}{4.2245}[/tex]
[tex]Z = 0.86[/tex]
[tex]Z = 0.86[/tex] has a p-value of 0.8051
0.8621 - 0.8051 = 0.057
The probability that exactly 27 of 104 eligible voters voted is 0.057 = 5.7%.
