Question

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.

Answer 1

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).