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

Answer:

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 $\pi$, and a confidence interval $1-\alpha$, we have the following confidence interval of proportions.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

Z is the zscore that has a pvalue of $1 - \frac{\alpha}{2}$.

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 $\pi = \frac{560}{1000} = 0.56$.

We have $\alpha = 0.05$, z is the value of Z that has a pvalue of $1 - \frac{0.05}{2}$ = 0.975, so $Z = 1.96$.

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.56 - 1.96\sqrt{\frac{0.56*0.44}{1000}} = 0.5292$

The upper limit of this interval is:

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

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.