Question

N a poll to estimate presidential popularity, each person in a random sample of 1,000 voters was asked to agree with one of the following statements: The president is doing a good job.The president is doing a poor job.I have no opinion.A total of 560 respondents selected the first statement, indicating they thought the president was doing a good job. (a) Construct a 95 percent confidence interval for the portion of respondents who feel the president is doing a good job.

(b) Based on your interval in part, is it reasonable to conclude that a majority (half) of the population believes the president is doing a good job?

Answer 1

a) The 95 percent confidence interval for the portion of respondents who feel the president is doing a good job is (0.5292, 0.5908).

b) The lower end of the confidence interval is above 0.5, so yes, it is reasonable to conclude that a majority (half) of the population believes the president is doing a good job.

Explanation:

The first step to solve this problem is building the confidence interval. If the lower end of the interval is above 0.5, it is reasonable to conclude that a majority of the population believes that the president is doing a good job.

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:

a)

A sample of 1000 voters was surveyed, and 560 feel that the president is doing a good job. This means that
`n = 1000` and
`\pi = (560)/(1000) = 0.56`.

We have
`\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)} = 0.56 - 1.96\sqrt{(0.56*0.44)/(1000)} = 0.5292`

The upper limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.56 + 1.96\sqrt{(0.56*0.44)/(1000)} = 0.5908`

The 95 percent confidence interval for the portion of respondents who feel the president is doing a good job is (0.5292, 0.5908).

b) The lower end of the confidence interval is above 0.5, so yes, it is reasonable to conclude that a majority (half) of the population believes the president is doing a good job.