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In order to determine how many people should be surveyed to estimate the proportion of votes for a candidate with a specified level of accuracy and confidence, it is essential to apply the principles of sampling theory and probability distributions. The situation described involves estimating a population proportion, which is the fraction of voters who support a particular candidate. Since it is usually impractical to survey the entire population, a sample is taken, and the sample proportion is used as an estimate of the true population proportion. The fundamental idea behind this problem is based on the sampling distribution of the sample proportion. When a sufficiently large random sample is drawn from a population, the sample proportion follows an approximately normal distribution due to the Central Limit Theorem. This approximation allows us to construct confidence intervals around the sample proportion and determine how close the estimate is likely to be to the true population value. The requirement is to be at least 95% confident that the estimate is within 0.05 of the true proportion. This means we are constructing a confidence interval with a margin of error equal to 0.05. The general formula for the margin of error when estimating a population proportion is given by: E = z \sqrt{\frac{p(1-p)}{n}} Here, (E) represents the margin of error, (z) is the critical value from the standard normal distribution corresponding to the desired ___ _______ _______ __________ ______ ________ ____ ________ ________ __________ _________ ________.
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