Confidence Interval

statistics

Definition

Confidence Interval

A confidence interval in statistics is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. The confidence level associated with the interval indicates the probability that the interval contains the parameter, typically expressed as a percentage like 95% or 99%.

It’s the range of plausible values.

Let $\mathbf{X}=(X_1,\dots ,X_n{)}^T$ be random sample. Then a confidence interval (a statistic) for the parameter $\theta$ with confidence level $1-\alpha$ is the random interval:

$${\mathrm{CI}}_{1-\alpha }(\mathbf{X})=[L(\mathbf{X}),R(\mathbf{X})]$$

Determined by the statistics (random variables) $L(\mathbf{X})$ and $R(\mathbf{X})$, such that:

$$\mathbb{P}[L(\mathbf{X})\le \theta \le R(\mathbf{X})]\ge 1-\alpha$$