Editing Confidence interval (section)

== Definition ==
Let <math>X</math> be a [[random sample]] from a [[probability distribution]] with [[statistical parameter]] <math>(\theta, \varphi)</math>. Here, <math>\theta</math> is the quantity to be estimated, while <math>\varphi</math> includes other parameters (if any) that determine the distribution. A confidence interval for the parameter <math>\theta</math>, with confidence level or coefficient <math>\gamma</math>, is an interval <math>(u(X), v(X))</math> determined by [[random variable]]s <math>u(X)</math> and <math>v(X)</math> with the property:
: <math>P(u(X) < \theta < v(X)) = \gamma \quad \text{for all }(\theta, \varphi).</math>

The number <math>\gamma</math>, whose typical value is close to but not greater than 1, is sometimes given in the form <math>1 - \alpha</math> (or as a percentage <math>100%\cdot(1 - \alpha)</math>), where <math>\alpha</math> is a small positive number, often 0.05. It means that the interval <math display="inline">(u(X), v(X))</math> has a probability <math display="inline">\gamma</math> of covering the value of <math display="inline">\theta</math> in repeated sampling. 

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 if

: <math>P(u(X) < \theta<v(X)) \approx\ \gamma</math>

to an acceptable level of approximation. Alternatively, some authors<ref>{{cite book |first=George G. |last=Roussas |year=1997 |title=A Course in Mathematical Statistics |edition=2nd |publisher=Academic Press |page=397}}</ref> simply require that

: <math>P(u(X) < \theta < v(X)) \ge\ \gamma</math>
When it is known that the [[coverage probability]] can be strictly larger than <math>\gamma</math> for some parameter values, the confidence interval is called conservative, i.e., it errs on the safe side; which also means that the interval can be wider than need be.
=== Methods of derivation ===
There are many ways of calculating confidence intervals, and the best method depends on the situation. Two widely applicable methods are [[Bootstrapping_(statistics)#Deriving_confidence_intervals_from_the_bootstrap_distribution|bootstrapping]] and the [[Central limit theorem|central limit theorem]].<ref name="Dekking">{{Cite journal |last1=Dekking |first1=Frederik Michel |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |title=A Modern Introduction to Probability and Statistics |url=https://link.springer.com/book/10.1007/1-84628-168-7 |journal=Springer Texts in Statistics |language=en-gb |doi=10.1007/1-84628-168-7 |isbn=978-1-85233-896-1 |issn=1431-875X}}</ref> The latter method works only if the sample is large, since it entails calculating the sample mean <math>\bar{X}_n</math> and sample standard deviation <math>S_n</math> and assuming that the quantity

: <math>\frac{\bar{X}_n - \mu}{S_n / \sqrt{n}}</math>

is normally distributed, where <math display="inline">\mu</math> and <math>n</math> are the population mean and the sample size, respectively.