Statistics for Business: Decision Making and Analysis, 3rd Edition

15.5 Margin of Error

The informal, back-of-the-envelope 95% confidence interval for μ replaces the percentile from the t-distribution with 2.

$$\left[\overline{x}-2s/\sqrt{n} \text{ to } \overline{x}+2s/\sqrt{n}\right]$$

Though informal, the extent of this interval to either side of $\overline{x}$ (or similarly around $\widehat{p}$) is known as the margin of error.

margin of error

$2\frac{s}{\sqrt{n}}$

$$\text{Margin} \text{of} \text{Error}=2\frac{s}{\sqrt{n}}$$

A precise confidence interval has a small margin of error. Three factors determine the margin of error.

1. Level of confidence. The multiplier 2 ≈ t_{0.025,n − 1} comes from choosing 95% coverage and using a t-distribution (or normal model) to describe the sampling distribution.

2. Variation of the data. The smaller the standard deviation s, the smaller the margin of error becomes. It’s usually not possible to reduce the standard ...