Large-sample intervals
Section of Estimation examples, part of Data analysis
Large-sample intervals
| Type | Section |
| Taught in | Estimation examples |
| Part | Data analysis |
When the data are not known to be normal, the central limit theorem still makes \overline X approximately normal for a large sample, and s is close to \sigma.
Definition 1 (Margin of error) 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.
The approximation needs care when the sample is small or the data are heavily skewed, which is what a histogram or a normal quantile plot checks. For a household fuel survey with n=51, \overline x=1136 and s=178 dollars, the approximate 95% interval is 1136\pm1.96\times178/\sqrt{51}=(1087,\ 1185). The exact normal-theory interval, with t_{50;0.975}=2.009, is (1086,\ 1186); at n=51 the two hardly differ.
The margin of error shrinks like 1/\sqrt n, so planning a sample size is a matter of solving for n: halving the margin takes four times the data. The calculation is in Confidence intervals.
Used in
See also
Margin of error · Welch interval · From data to an estimate · Small normal samples · Two samples with unequal variances · Intervals for proportions · Wilcoxon rank-sum test