Meaning of margin of error calculation
In estimation, a margin of error calculation produces a nonnegative number \(E\) such that a confidence interval has the form \[ \text{estimate} \pm E. \] For a population mean with known \(\sigma\), the interval is centered at \(\bar{x}\) and the margin of error depends on a critical value and the standard error.
Known-\(\sigma\) mean (z interval)
\[ E = z_{\alpha/2}\cdot \frac{\sigma}{\sqrt{n}}, \qquad \text{CI for }\mu:\ \bar{x} \pm E. \]
Here \(\alpha=1-\text{confidence level}\), and \(z_{\alpha/2}\) is the standard normal critical value with upper-tail area \(\alpha/2\).
Step-by-step solution for the given data
Given \(n=64\), \(\bar{x}=52.4\), \(\sigma=10\), and 95% confidence: \[ \alpha = 1-0.95 = 0.05, \quad \alpha/2=0.025, \quad z_{\alpha/2}=z_{0.025}=1.96. \]
1) Compute the standard error
\[ \frac{\sigma}{\sqrt{n}}=\frac{10}{\sqrt{64}}=\frac{10}{8}=1.25. \]
2) Margin of error calculation
\[ E=z_{\alpha/2}\cdot \frac{\sigma}{\sqrt{n}} =1.96\cdot 1.25 =2.45. \]
3) Confidence interval
\[ \bar{x}\pm E = 52.4 \pm 2.45. \] Lower endpoint: \[ 52.4 - 2.45 = 49.95. \] Upper endpoint: \[ 52.4 + 2.45 = 54.85. \]
Result
Margin of error: \(E=2.45\).
95% confidence interval for \(\mu\): \((49.95,\ 54.85)\).
Visualization: estimate \(\pm E\) on a number line
The center mark is \(\bar{x}=52.4\). The endpoints are \(\bar{x}-E=49.95\) and \(\bar{x}+E=54.85\). The bracket length corresponds to \(2E\).
Reference: common critical values for margin of error calculation
For a two-sided z-based interval (known \(\sigma\)), \(z_{\alpha/2}\) depends only on the confidence level.
| Confidence level | \(\alpha\) | \(\alpha/2\) | \(z_{\alpha/2}\) |
|---|---|---|---|
| 90% | 0.10 | 0.05 | 1.645 |
| 95% | 0.05 | 0.025 | 1.96 |
| 99% | 0.01 | 0.005 | 2.576 |
Extensions frequently needed in practice
Mean with \(\sigma\) unknown (t interval)
When \(\sigma\) is unknown, replace \(\sigma\) by sample standard deviation \(s\) and use a t critical value.
\[ E = t_{\alpha/2,\,n-1}\cdot \frac{s}{\sqrt{n}}, \qquad \text{CI for }\mu:\ \bar{x}\pm E. \]
Population proportion (large-sample z interval)
For \(\hat{p}\) with sufficiently large counts, the margin of error uses the standard error of \(\hat{p}\).
\[ E = z_{\alpha/2}\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \qquad \text{CI for }p:\ \hat{p}\pm E. \]
Sample size planning from a target margin of error
- Mean (known \(\sigma\)): to achieve margin \(E\), \[ n \ge \left(\frac{z_{\alpha/2}\cdot \sigma}{E}\right)^2. \]
- Proportion: to achieve margin \(E\), \[ n \ge \frac{z_{\alpha/2}^2\cdot \hat{p}(1-\hat{p})}{E^2}, \] and the conservative choice \(\hat{p}=0.5\) maximizes \(\hat{p}(1-\hat{p})\).