Margin of error
Definition taught in Estimation examples, part of Data analysis
Margin of error
| Type | Definition |
| Taught in | Estimation examples |
| Part | Data analysis |
| Worked examples | 2 |
| Questions using it | 3 |
For a random sample with mean \mu and finite variance, the approximate confidence interval for \mu with confidence 1-\alpha is \overline X\pm z_{1-\alpha/2}\,s/\sqrt n. Its half-width, z_{1-\alpha/2}\,s/\sqrt n, is the margin of error.
Examples
Narrowing a pilot interval
Question page: Narrowing a pilot interval
A pilot sample of 100 elderly people in nursing homes gave a 95% confidence interval (59.3,\,82.5) kg for the mean weight. What sample size is needed for a 95% margin of error of 1 kg?
The pilot half-width is (82.5-59.3)/2=11.6 kg at n=100. The margin scales as 1/\sqrt n, so a margin of 1 kg needs
n\ge 100\times 11.6^2=13456.
Recovering the pilot standard deviation with a t_{99} quantile near 1.98 and then planning the new interval with z_{0.975}=1.96 lowers the requirement to 13188. Either figure is a sample of more than thirteen thousand.
Hours of sleep
Question page: Hours of sleep
Four students produced a 95% confidence interval 16.753\pm 7.39 hours for the mean hours slept over a weekend. What is the smallest sample size that would bring the margin of error down to 2 hours, keeping the same estimated standard deviation and the same normal approximation to the critical value?
The pilot margin is 7.39=t_{3,\,0.975}\,s/\sqrt{4} with t_{3,\,0.975}\approx 3.182, so s\approx 4.65. A future sample of size n has margin t_{n-1,\,0.975}\,s/\sqrt n. The smallest n for which this is at most 2 is n=24: t_{23,\,0.975}\approx 2.07 and 2.07\times 4.65/\sqrt{24}\approx 1.96<2, while n=23 is just over 2.
Used in
- Taught in Estimation examples, where the derivation and examples are.
- Questions that use it (3) — Narrowing a pilot interval · Hours of sleep · How far sharks swim
- All 7 questions on Estimation examples