Question

A survey asked whether respondents favored or opposed the death penalty for people convicted of murder. Software shows the results​ below, where X refers to the number of the respondents who were in favor. Construct the​ 95% confidence interval for the proportion of the adults who were opposed to the death penalty from the confidence interval stated below for the proportion in favor.

X N Sample p ​ 95.0% CI
1786 2611 0.684 (0.666,0.702 )
The​ 95% confidence interval for those opposed is:___________ ​ (.666 . 666​, .702 . 702​).

Answer 1

The 95% confidence interval for those opposed is: (0.298, 0.334).

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:

1786 of the 2611 were in favor, so 2611 - 1786 = 825 were opposed. Then

`n = 2611, \pi = (825)/(2611) = 0.316`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.316 - 1.96\sqrt{(0.316*0.684)/(2611)} = 0.298`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.316 + 1.96\sqrt{(0.316*0.684)/(2611)} = 0.334`

The 95% confidence interval for those opposed is: (0.298, 0.334).