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During a local election between two candidates, exit polls based on a sample of 400 voters indicated that 54% of the voters supported the incumbent candidate. Construct a 90% confidence interval for the percentage of votes that the incumbent has received in this election.

User Dudette
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Answer:

The 90% confidence interval for the percentage of votes that the incumbent has received in this election is (0.499, 0.581).

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
\pi, and a confidence level of
1-\alpha, we have the following confidence interval of proportions.


\pi \pm z\sqrt{(\pi(1-\pi))/(n)}

In which

z is the zscore that has a pvalue of
1 - (\alpha)/(2).

For this problem, we have that:


n = 400, p = 0.54

90% confidence level

So
\alpha = 0.1, z is the value of Z that has a pvalue of
1 - (0.1)/(2) = 0.95, so
Z = 1.645.

The lower limit of this interval is:


\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.54 - 1.645\sqrt{(0.54*0.46)/(400)} = 0.499

The upper limit of this interval is:


\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.54 + 1.645\sqrt{(0.54*0.46)/(400)} = 0.581

The 90% confidence interval for the percentage of votes that the incumbent has received in this election is (0.499, 0.581).

User Aleksander Grzyb
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