Question

In 2015 as part of the General Social Survey, 1289 randomly selected American adults responded to this question:

"Some countries are doing more to protect the environment than other countries. In general, do you think that America is doing more than enough, about the right amount, or too little?"

Of the respondents, 436 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.

A (0.304, 0.372)

B (0.312, 0.364)

C (0.325, 0.351)

D (0.317, 0.36)

Answer 1

B (0.312, 0.364)

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence interval
`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:

1289 randomly selected American adults responded to this question. This means that
`n = 1289`.

Of the respondents, 436 replied that America is doing about the right amount. This means that
`\pi = (436)/(1289) = 0.3382`.

Determine a 95% confidence interval for the proportion of all the registered voters who will vote for the Republican Party. ​

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)} = 0.3382 - 1.96\sqrt{(0.3382*0.6618)/(1289)} = 0.312`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.3382 + 1.96\sqrt{(0.3382*0.6618)/(1289)} = 0.364`

The 95% confidence interval is:

B (0.312, 0.364)