Use the formula for the standard error of p̂ to explain why a. the standard error is greater whe

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Use the formula for the standard error of p̂ to explain why...

Use the formula for the standard error of $\hat{p}$ to explain why a. the standard error is greater when the value of the population proportion $p$ is near 0.5 than when it is near 1 . b. the standard error of $\hat{p}$ is the same when the value of the population proportion is $p=0.2$ as it is when $p=0.8$.

Use the formula for the standard error of $\hat{p}$ to explain why
a. the standard error is greater when the value of the population proportion $p$ is near 0.5 than when it is near 1 .
b. the standard error of $\hat{p}$ is the same when the value of the population proportion is $p=0.2$ as it is when $p=0.8$.

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Key Concepts

Standard Error of a Proportion

This concept refers to the measure of variability in the sample proportion, typically calculated using the formula SE = ?(p(1-p)/n). It quantifies how much the sample proportion is expected to fluctuate around the true population proportion due to random sampling error.

Maximum Variability at p = 0.5

The product p(1-p) in the standard error formula reaches its maximum when p equals 0.5 because this is the point where the two factors are balanced. This maximum value leads to the highest standard error, indicating that sample proportions are most variable when the true population proportion is around 0.5.

Symmetry of the p(1-p) Function

The function p(1-p) is symmetric about p = 0.5, meaning that p(1-p) has the same value for complementary probabilities (e.g., 0.2 and 0.8). This symmetry explains why the standard error remains the same when the population proportion is 0.2 or 0.8, as the quality of variability induced by the product is identical for these complementary values.

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