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?
$p=.9$

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Key Concepts

Binomial Proportion

A binomial proportion is the probability of a specific outcome in a binary experiment, calculated as the number of successful outcomes divided by the number of trials. This concept is crucial when dealing with experiments or surveys where each trial has only two possible outcomes, like success or failure.

Margin of Error

The margin of error quantifies the uncertainty around an estimate from a sample, such as the binomial proportion. It is computed as a product of a critical value (from a relevant probability distribution) and the standard error of the proportion, which assesses the variability due to sampling.

Standard Error of a Proportion

The standard error of a proportion, given by the square root of [p(1-p)/n], measures the precision of the sample proportion as an estimate of the true population proportion. It depends on both the sample size and the true proportion, and plays a key role in constructing confidence intervals.

Maximum Variability (p = 0.5)

For a binomial distribution, the product p(1-p) reaches its maximum at p = 0.5. This maximum variability results in the largest standard error and, consequently, the largest margin of error in estimating the proportion. Therefore, p = 0.5 produces the greatest uncertainty in the estimate.

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