Confidence interval

In frequentist inference, a confidence interval (CI) determines lower and upper bounds likely to contain (in repeated sampling) the true value of an unknown statistical parameter, such as a population mean. Rather than reporting a single point estimate (e.g. "the average screen time is 3 hours per day"), a confidence interval provides an interval, such as from 2 hours to 4 hours.

A confidence interval is based on a pre-defined coverage probability, called confidence level, typically 95%. A 95% confidence level does not imply a 95% probability that the true parameter lies within a particular CI calculated from a sample (such interpretation would be associated with the credible interval in Bayesian inference). The confidence level instead reflects the long-run performance of the method used to generate the interval. In other words, if the same sampling procedure were repeated 100 times from the same population, approximately 95 of the resulting intervals would be expected to contain the true population mean. The frequentist approach sees the true population mean as a fixed unknown constant, while the confidence interval is calculated using data from a random sample. Because the sample is random, the interval endpoints are random variables.

Definition

Let

X

{\displaystyle X}

be a random sample from a probability distribution with statistical parameter

(

θ

,

φ

)

{\displaystyle (\theta ,\varphi )}

.