A researcher wants to estimate the proportion of students...

A researcher wants to estimate the proportion of students enrolled at a university who are registered to vote. Would the standard error of the sample proportion $\hat{p}$ be larger if the actual population proportion was $p=0.4$ or $p=0.8 ?$

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Key Concepts

Standard Error of a Proportion

This concept measures the variability of the sample proportion from one sample to another. It's calculated using the formula sqrt[p(1-p)/n], where p is the true population proportion and n is the sample size. This formula shows that the standard error depends on both the population proportion and the sample size, and that variability is higher when p(1-p) is larger.

Sampling Distribution of the Sample Proportion

This refers to the probability distribution of the sample proportion obtained from repeated random samples of a fixed size from the population. Under certain conditions, particularly with a large sample size, this distribution is approximately normal, and its spread is quantified by the standard error. Understanding this distribution is crucial for making inferences about the population proportion.

Impact of the Population Proportion on Variability

The term p(1-p) within the standard error formula indicates that the variability of the sample proportion is not constant but depends on the actual value of p. It reaches its maximum when p is 0.5 and diminishes as p moves closer to 0 or 1, meaning that for proportions further away from 0.5, like p = 0.8, the standard error is smaller than for proportions closer to 0.5, such as p = 0.4.

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