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yuradex [85]
3 years ago
7

3. A recent news article stated that only 17% of college students between the ages of 18 to 24 years old voted in the last presi

dential election. Assuming the voting rate stays the same, what is the probability that from a random sample of 500 college students from a local university, at least 20% will vote in the next presidential election
Mathematics
1 answer:
dangina [55]3 years ago
5 0

Answer:

3.67% probability that from a random sample of 500 college students from a local university, at least 20% will vote in the next presidential election

Step-by-step explanation:

I am going to use the binomial approximation to the normal to solve this question.

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 samples 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 problem, we have that:

p = 0.17, n = 500.

So

\mu = E(X) = np = 500*0.17 = 85

\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{500*0.17*0.83} = 8.4

Assuming the voting rate stays the same, what is the probability that from a random sample of 500 college students from a local university, at least 20% will vote in the next presidential election

This is 1 subtracted by the pvalue of Z when X = 500*0.2 = 100. So

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

Z = \frac{100 - 85}{8.4}

Z = 1.79

Z = 1.79 has a pvalue of 0.9633

1 - 0.9633 = 0.0367

3.67% probability that from a random sample of 500 college students from a local university, at least 20% will vote in the next presidential election

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