A sufficient number of voters are polled to determine the...

A sufficient number of voters are polled to determine the percentage in favor of a certain candidate. Assuming that an unknown proportion $p$ of the voters favor him and they act independently of one another, how many should be polled to predict the value of $p$ within $4.5 \%$ with $95 \%$ confidence? [This is the so-called four percent margin of error in predicting elections, presumably because $<.045$ becomes $\leq .04$ by the rule of rounding decimals.

A sufficient number of voters are polled to determine the percentage in favor of a certain candidate. Assuming that an unknown proportion $p$ of the voters favor him and they act independently of one another,
how many should be polled to predict the value of $p$ within $4.5 \%$ with $95 \%$ confidence? [This is the so-called four percent margin of error in predicting elections, presumably because $<.045$ becomes $\leq .04$ by the rule of rounding decimals.

Key Concepts

Confidence Interval for Proportions

This is a statistical interval estimate that is computed from sample data and is likely to contain the true population proportion within a specified level of confidence. The concept combines sample data and the idea of variability to give a range of plausible values for the unknown proportion, which is especially useful in scenarios like election polling.

Margin of Error

The margin of error quantifies the maximum expected difference between the true population parameter and the sample estimate due to random sampling variability. In the context of estimating a population proportion, it is determined by the critical value from the normal distribution, the sample variance, and the size of the sample, and it represents the precision of the estimate.

Normal Approximation and Z-Score

When the sample size is sufficiently large, the sampling distribution of the proportion can be approximated by a normal distribution according to the Central Limit Theorem. The Z-score, corresponding to the desired confidence level (e.g., 1.96 for 95% confidence), is used to scale the standard error and thereby determine the margin of error for the confidence interval.

Sample Size Calculation

Determining the required sample size involves calculating how many observations are needed to achieve a specified margin of error at a given confidence level. For proportions, this calculation typically uses the formula n = (Z^2 * p*(1-p))/(E^2), where p*(1-p) is maximized at 0.25 when p is unknown, Z is the critical value, and E is the desired margin of error. This ensures that the estimate of the proportion is sufficiently precise for practical use.

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