Question

A poll is taken in which 304 out of 600 randomly selected voters indicated their preference for a certain candidate. (a) Find a 95% confidence interval for p.

__≤ p ≤ __

Answer 1

The 95% confidence interval for the population proportion of voters who favor a certain candidate, based on a sample of 304 out of 600 voters, is calculated to be between 46.74% and 54.60%.

Step-by-step explanation:

The subject of this question is Mathematics, specifically statistics. The student is asking for a 95% confidence interval for the population proportion (p) based on a sample where 304 out of 600 voters favor a certain candidate. To find this confidence interval, we need to use the formula for the confidence interval for a population proportion which is:

p' ± Z* √ [(p'(1 - p')) / n]

where p' is the sample proportion, Z* is the Z-score corresponding to the desired confidence level, and n is the sample size.

Step-by-Step Calculation

1. Calculate the sample proportion (p'): p' = 304/600 = 0.5067.
2. Determine the Z-score for a 95% confidence interval: Z* = 1.96 (This value comes from the standard normal distribution table and corresponds to the 97.5th percentile, as 2.5% is in each tail).
3. Compute the margin of error (EBP): EBP = 1.96 √ [(0.5067*(1 - 0.5067)) / 600] = 0.0393.
4. Calculate the confidence interval: Lower limit = p' - EBP = 0.5067 - 0.0393 = 0.4674, Upper limit = p' + EBP = 0.5067 + 0.0393 = 0.5460.

Therefore, we are 95% confident that the true proportion of voters who favor the candidate is between 46.74% and 54.60%.