Why do polling companies often survey 1060 individuals when they wish to estimate a population proportion with a margin of error of 3% with 95% confidence?

Verified step by step guidance

1. Understand that the margin of error (ME) for estimating a population proportion \( p \) with a confidence level is given by the formula: \[ ME = z^{*} \times \sqrt{\frac{p(1-p)}{n}} \] where \( z^{*} \) is the critical value from the standard normal distribution corresponding to the desired confidence level, \( p \) is the estimated population proportion, and \( n \) is the sample size.
2. Identify the critical value \( z^{*} \) for a 95% confidence level. This value corresponds to the point where the central 95% of the normal distribution lies between \( -z^{*} \) and \( z^{*} \).
3. Recognize that when the true population proportion \( p \) is unknown, polling companies often use \( p = 0.5 \) because it maximizes the product \( p(1-p) \), which leads to the largest required sample size and thus ensures the margin of error will not be underestimated.
4. Rearrange the margin of error formula to solve for the sample size \( n \): \[ n = \frac{(z^{*})^{2} \times p(1-p)}{ME^{2}} \] Here, \( ME = 0.03 \) (3%), \( p = 0.5 \), and \( z^{*} \) corresponds to 95% confidence.
5. Calculate the sample size \( n \) using the values above to find why approximately 1060 individuals are surveyed. This ensures the margin of error is within 3% at 95% confidence.

Key Concepts

Margin of Error

The margin of error quantifies the range within which the true population parameter is expected to lie, given a certain confidence level. It reflects the maximum expected difference between the sample estimate and the actual population value, often expressed as a percentage.

Confidence Level

The confidence level represents the probability that the margin of error contains the true population parameter. A 95% confidence level means that if the survey were repeated many times, 95% of the intervals would include the true proportion.

Sample Size Determination for Proportions

Sample size is calculated to achieve a desired margin of error and confidence level when estimating a population proportion. It depends on the variability of the proportion and the z-score corresponding to the confidence level, ensuring the estimate is precise enough.