Calculate the margin of error in estimating a binomial...

Calculate the margin of error in estimating a binomial proportion $p$ using samples of size $n=100$ and the values of p given in Exercises $15-19 .$ What value of p produces the largest margin of error?
$p=.1$

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Key Concepts

Margin of Error

The margin of error is a measure of the uncertainty in the estimate of a parameter, typically calculated for a confidence interval. In the context of estimating a proportion, it is determined by the variability of the estimate and decreases as the sample size increases, thereby providing a range within which the true parameter is likely to lie.

Standard Error of a Proportion

The standard error for a binomial proportion is given by the formula ?(p(1?p)/n), illustrating how the variability of the estimated proportion depends on both the true proportion p and the sample size n. This measure is essential because it directly influences the margin of error in the estimation.

Binomial Proportion

A binomial proportion refers to the probability p of a success in a series of independent Bernoulli trials. It represents the fraction of successes in a sample and is a key parameter in binomial distribution analysis, often estimated from sample data.

Maximum Uncertainty Condition

The product p(1?p) reaches its maximum value when p is 0.5, which means that the variability—and consequently the margin of error—is largest at this point. This concept is crucial as it identifies the worst-case scenario for the estimation error of the proportion.

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