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Concept
Confidence interval
Summary
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated confidence level; the 95% confidence level is most common, but other levels, such as 90% or 99%, are sometimes used. The confidence level, degree of confidence or confidence coefficient represents the long-run proportion of CIs (at the given confidence level) that theoretically contain the true value of the parameter; this is tantamount to the nominal coverage probability. For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter's true value.
Factors affecting the width of the CI include the sample size, the variability in the sample, and the confidence level. All else being the same, a larger sample produces a narrower confidence interval, greater variability in the sample produces a wider confidence interval, and a higher confidence level produces a wider confidence interval.
Let be a random sample from a probability distribution with statistical parameter , which is a quantity to be estimated, and , representing quantities that are not of immediate interest. A confidence interval for the parameter , with confidence level or coefficient , is an interval determined by random variables and with the property:
The number , whose typical value is close to but not greater than 1, is sometimes given in the form (or as a percentage ), where is a small positive number, often 0.05.
It is important for the bounds and to be specified in such a way that as long as is collected randomly, every time we compute a confidence interval, there is probability that it would contain , the true value of the parameter being estimated. This should hold true for any actual and .
In many applications, confidence intervals that have exactly the required confidence level are hard to construct, but approximate intervals can be computed. The rule for constructing the interval may be accepted as providing a confidence interval at level if
to an acceptable level of approximation.
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