Question

Political science, inc. (psi) specializes in voter polls and surveys designed to keep political office seekers informed of their position in a race. Using telephone surveys, interviewers ask registered voters who they would vote for if the election were held that day. ​In a recent election campaign, psi found that 220 registered voters, out of 500 contacted, favored a particular candidate. ​

a. develop a 95% confidence interval estimate for the proportion of the population of registered voters that favors the candidate. Interpret the interval.
b. suppose that psi would like 99% confidence that the sample proportion is within ± .03 of the population proportion. What sample size is needed to provide the desired margin of error?

Answer 1

A 95% confidence interval for the proportion of the population that favors the candidate is between 39.68% and 48.32%. For a 99% confidence level with a margin of error of ± .03, PSI needs to sample approximately 1,515 registered voters.

Step-by-step explanation:

Part A: Confidence Interval Estimate

To develop a 95% confidence interval for the proportion of the population that favors the candidate, we use the formula for a confidence interval of a proportion:

CI = p ± Z*(√(p(1-p)/n))

Where p is the sample proportion, n is the sample size, and Z is the z-score corresponding to the desired confidence level. In this case, p = 220/500 = 0.44, n = 500, and Z for a 95% confidence interval is approximately 1.96.

The confidence interval is: 0.44 ± 1.96*(√(0.44(0.56)/500))

Calculating the margin of error, we find the interval to be:

0.44 ± 0.0432

This interval can be interpreted to mean that we are 95% confident that the true proportion of registered voters that favors the candidate lies between 39.68% and 48.32%.

Part B: Determining Necessary Sample Size

To determine the sample size needed to have a margin of error of ± .03 with a 99% confidence level, the following formula for sample size is used:

n = (Z^2 · p(1-p)) / E^2

Z corresponding to 99% confidence is approximately 2.576, p is the estimated proportion (we can use 0.44 from the previous sample unless we have a better estimate), and E is the desired margin of error which is 0.03.

n = (2.576^2 · 0.44 · 0.56) / 0.03^2

Calculating this gives us a required sample size of approximately:

n = 1,515. This is the number of registered voters PSI would need to contact to achieve the desired margin of error with 99% confidence.