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Confidence Interval

The confidence interval is said to be the particular kind of the interval estimate of a definite population parameter which is used for indicating the reliability of any estimate in the subject of statistics. It is calculated on the basis of observations and is thus an observed interval which varies in principle from sample to sample and also includes the parameters of interest, if the experiments are repeated. The frequency of the observed sample containing the various parameters is determined by the confidence coefficient or the confidence level. If the statistical model is correct, then the confidence interval (which has a particular confidence level) is used for giving an assurance that the procedure when undertaken for constructing the interval will also deliver a confidence interval which will include a true value of the parameter that will be in proportion of the time which is set by the confidence levels. Confidence levels more specifically means that when the confidence intervals are constructed across various data analysis programs in repeated and different experiments, the proportions of all these intervals contain the true values of the parameters and also approximately match the confidence level.

The interval estimates can be seen in contrast with the point estimates. A single value which is given by estimating a population parameter and which is of interest is called point estimate. In contrast, the interval estimate is used to specify a range in which the parameter is estimated to lie. The tables and graphs commonly contain the reported confidence intervals which are reported along with the point estimates containing the same parameters, in order to show the reliability of estimates.

Confidence region – is used for generalizing the confidence interval concept in order to deal with the multiple quantities. These regions help in indicating the extent to which the sampling errors can occur and also reveal the estimates for the reliability and unreliability of the quantities and their relationship. When practically applied, the confidence levels are stated at 95%confidence levels and when presented in graphs, they can be shown at various confidence levels, for example at 50%, 95% or 99%.

Definition of confidence interval

When X is a random sample in a probability distribution having parameter θ as the quantity being estimated and φ representing the quantities which are not of immediate interest, then the confidence interval for the parameter θ having the confidence coefficient γ is an interval having random endpoints which are determined by the pair of statistics or the observable random variables u(X) and v(X), having the property



Here, the quantities of φ for which there are no immediate interests are called nuisance parameters. The number γ is also given the sometimes of 1 − α, and has typical values which are close to, but not greater than 1.

Desirable properties in confidence intervals include.

Validity- It means that the nominal coverage probability or the confidence level of the confidence must hold exactly or to possible good approximation level.

Optimality- The information contained in the data set must be used to highest extent while constructing the confidence interval.

Invariance- The methods which are used for constructing the confidence interval must be able to give equivalent results.

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