Question

Of the respondents, 502 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?

Answer 1

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 is (0.461, 0.543), considering
`n = 1000`

Explanation:

Incomplete question, so i will suppose this is a sample of 1000.

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 z-score that has a p-value of
`1 - (\alpha)/(2)`.

Of the n respondents, 502 replied that America is doing about the right amount.

Supposing
`n = 1000`, so
`\pi = (502)/(1000) = 0.502`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a p-value 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)} = 0.502 - 2.575\sqrt{(0.502*0.498)/(1000)} = 0.461`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.502 + 2.575\sqrt{(0.502*0.498)/(1000)} = 0.543`

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 is (0.461, 0.543), considering
`n = 1000`