Question

In 2010 as part of the General Social Survey, 1295 randomly selected American adults responded to this question: "Some countries are doing more to protect the environment than other countries are. In general, do you think that America is doing more than enough, about the right amount, or too little?" Of the respondents, 473 replied that America is doing about the right amount. What is the 95% confidence interval for the proportion of all American adults who feel that America is doing about the right amount to protect the environment. Group of answer choices (0.352, 0.378) (0.343, 0.387) (0.339, 0.391) (0.331, 0.400)

Answer 1

(0.339, 0.391)

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 = 1295, \pi = (473)/(1295) = 0.365`

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)} = 0.365 - 1.96\sqrt{(0.365*0.635)/(1295)} = 0.339`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.365 + 1.96\sqrt{(0.365*0.635)/(1295)} = 0.391`

So the correct answer is:

(0.339, 0.391)