Na 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?

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.

Step-by-step 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 , and a confidence interval , we have the following confidence interval of proportions.

In which

Z is the zscore that has a pvalue of .

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 and .

We have , z is the value of Z that has a pvalue of = 0.975, so .

The lower limit of this interval is:

The upper limit of this interval is:

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.

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