Mathematics, 19.05.2021 09:00 clonetrooper099

Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 104 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted. Probability that exactly 27 voted

The probability that exactly 27 of 104 eligible voters voted is

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Answer from: kukisbae

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:

E(X) = np

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $\mu = E(X)$ , $\sigma = \sqrt{V(X)}$ .

In this case, assume that 104 eligible voters aged 18-24 are randomly selected.

This means that n = 104 .

Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted.

This means that p = 0.22

Mean and standard deviation:

$\mu = 104*0.22 = 22.88$

$\sigma = \sqrt{104*0.22*0.78} = 4.2245$

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

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{27.5 - 22.88}{4.2245}$

Z = 1.09

Z = 1.09 has a p-value of 0.8621

X = 26.5

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{26.5 - 22.88}{4.2245}$

Z = 0.86

Z = 0.86 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%.

Answer from: Quest

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step-by-step explanation:

Answer from: Quest

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