Margin of error Knowledge-Based Study

Study notes

1. What is Margin of Error?

   In statistics, researchers rarely collect data from an entire population. Instead, they select a representative sample to estimate unknown population parameters, such as a population mean or a population proportion. Because different samples produce slightly different estimates, a sample statistic will almost never match the true population parameter exactly. This discrepancy is known as sampling variability.

   The margin of error (MOE) quantifies the maximum expected difference between the sample estimate and the true population parameter at a specified level of confidence. It defines the precision of a sample estimate by establishing a boundary above and below the point estimate.

   Formulas for Common Estimates

   The calculation of the margin of error depends on whether the parameter of interest is a proportion or a mean.

   1. Margin of Error for a Population Proportion

   When estimating a population proportion $p$ using a sample proportion $\hat{p}$, the margin of error is calculated as:

   $$MOE = z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$$

   where $z^*$ is the critical value corresponding to the chosen confidence level, $\hat{p}$ is the sample proportion, and $n$ is the sample size.

   2. Margin of Error for a Population Mean

   When estimating a population mean $\mu$ with a known population standard deviation $\sigma$, the formula is:

   $$MOE = z^* \frac{\sigma}{\sqrt{n}}$$

   If the population standard deviation is unknown and estimated using the sample standard deviation $s$, the $t$-distribution is used instead:

   $$MOE = t^* \frac{s}{\sqrt{n}}$$

   Key Determinants of the Margin of Error

   Three main factors determine the size of the margin of error:

   • Sample Size ($n$): Because $n$ is in the denominator under a square root, increasing the sample size decreases the margin of error. To cut the margin of error in half, the sample size must be quadrupled.
   • Confidence Level ($z^*$ or $t^*$): A higher confidence level requires a larger critical value. For instance, a 95% confidence level uses $z^* \approx 1.960$, whereas a 99% confidence level uses $z^* \approx 2.576$. Increasing the confidence level increases the margin of error, making the interval wider.
   • Variability in the Data: Greater population variability (larger $\sigma$ or $s$) increases standard error, resulting in a larger margin of error. For proportions, variability is maximized when $\hat{p} = 0.5$.

   Interpreting Confidence Intervals and Caveats

   A point estimate combined with its margin of error forms a confidence interval:

   $$\text{Confidence Interval} = \text{Point Estimate} \pm MOE$$

   For example, if a survey finds that 58% of respondents support a proposal with a margin of error of $\pm 3\%$ at a 95% confidence level, the resulting interval is from 55% to 61%. This means we are 95% confident that the true population proportion of supporters lies between 0.55 and 0.61.

   Crucial Limitation

   The margin of error accounts only for random sampling error. It does not account for systematic errors such as selection bias, non-response bias, poorly worded survey questions, or measurement error.