Understanding the Standard Error of the Proportion

The standard error of the proportion estimates how much a sample proportion \( \hat{p} \) would vary if multiple samples were drawn from the same population. It’s used to construct confidence intervals and test hypotheses about proportions.

Formula for Standard Error of the Proportion

The standard error \( SE \) is calculated using the following formula:

$$ SE = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} $$

where:

• \( \hat{p} \): Sample proportion
• \( n \): Sample size

Example Calculation

Suppose a survey finds that 60% of respondents support a new policy. With a sample size of 150, the standard error is calculated as follows:

• Step 1: Set \( \hat{p} = 0.6 \) and \( n = 150 \).
• Step 2: Substitute values into the formula: \[ SE = \sqrt{\frac{0.6 \times (1 - 0.6)}{150}} = 0.039 \]

This means the standard error of the proportion is approximately 0.039, giving an idea of the sampling variability.