**Definition**

Let *X* be a random sample from a probability distribution with statistical parameters *θ*, which is a quantity to be estimated, and *φ*, representing quantities that are not of immediate interest. A *confidence interval* for the parameter *θ*, with confidence level or confidence coefficient *γ*, is an interval with random endpoints (*u*(*X*), *v*(*X*)), determined by the pair of random variables *u*(*X*) and *v*(*X*), with the property:

The quantities *φ* in which there is no immediate interest are called nuisance parameters, as statistical theory still needs to find some way to deal with them. The number *γ*, with typical values close to but not greater than 1, is sometimes given in the form 1 − *α* (or as a percentage 100%·(1 − *α*)), where *α* is a small non-negative number, close to 0.

Here Pr_{θ,φ} indicates the probability distribution of *X* characterised by (*θ*, *φ*). An important part of this specification is that the random interval (*u*(*X*), *v*(*X*)) covers the unknown value *θ* with a high probability no matter what the true value of *θ* actually is.

Note that here Pr_{θ,φ} need not refer to an explicitly given parameterised family of distributions, although it often does. Just as the random variable *X* notionally corresponds to other possible realizations of *x* from the same population or from the same version of reality, the parameters (*θ*, *φ*) indicate that we need to consider other versions of reality in which the distribution of *X* might have different characteristics.

In a specific situation, when *x* is the outcome of the sample *X*, the interval (*u*(*x*), *v*(*x*)) is also referred to as a confidence interval for *θ*. Note that it is no longer possible to say that the (observed) interval (*u*(*x*), *v*(*x*)) has probability *γ* to contain the parameter *θ*. This observed interval is just one realization of all possible intervals for which the probability statement holds.

Read more about this topic: Confidence Interval, Statistical Theory

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