Death Penalty In a Gallup poll conducted in October 2010,64...

Death Penalty In a Gallup poll conducted in October $2010,64 \%$ of the people polled answered yes to the following question: "Are you in favor of the death penalty for a person convicted of murder?"The margin of error in the poll was $3 \%,$ and the estimate was made with $95 \%$ confidence. At least how many people were surveyed?

Key Concepts

Margin of Error

The margin of error indicates the maximum expected difference between the true population parameter and a sample estimate. It reflects the precision of the survey results, meaning that the true proportion in the population is expected to lie within this range above or below the estimated proportion with a certain level of confidence.

Confidence Level

The confidence level represents the degree of certainty that the true population parameter lies within the specified margin of error. For example, a 95% confidence level means that if the same survey were conducted multiple times, approximately 95% of the time the true parameter would fall within the calculated interval.

Z-Score (Critical Value)

The z-score or critical value corresponds to the number of standard deviations a data point is from the mean, based on a standard normal distribution. In the context of confidence intervals, the critical value (approximately 1.96 for a 95% confidence level) is used to scale the standard error to achieve the desired level of confidence.

Sample Size Calculation for Proportions

Determining the required sample size for a survey estimating a proportion involves applying a formula that relates the desired margin of error, the confidence level (and its corresponding z-score), and the estimated proportion. The formula n = (z^2 * p * (1-p)) / (MOE^2) allows one to calculate the minimum number of respondents needed to achieve a specified precision in the estimate.

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