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?
$p=.3$

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Key Concepts

Margin of Error

The margin of error quantifies the expected range of error when estimating a population parameter from a sample. In the context of estimating a binomial proportion, it is typically derived by multiplying the standard error of the sample proportion by the critical value (such as a z-score) corresponding to the desired confidence level.

Standard Error of a Proportion

The standard error for a binomial proportion is calculated as the square root of [p(1 - p)/n], where p represents the proportion of success and n is the sample size. This measure indicates the degree of variability of the sample proportion around the true population proportion.

Maximum Variability in Binomial Proportions

The variability in estimating a binomial proportion reaches its maximum when p(1 - p) is maximized. Since the function p(1 - p) attains its highest value at p = 0.5, this is the proportion that produces the largest margin of error. This concept is crucial when designing studies to ensure that the sample size is adequate for the level of precision desired.

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