Confidence interval

In statistics, a confidence interval (CI) is a type of estimate computed from the statistics of the observed data. This gives a range of values for an unknown parameter (for example, a population mean). The interval has an associated confidence level that gives the probability with which an estimated interval will contain the true value of the parameter. The confidence level is chosen by the investigator. For a given estimation in a given sample, using a higher confidence level generates a wider (i.e., less precise) confidence interval. In general terms, a confidence interval for an unknown parameter is based on sampling the distribution of a corresponding estimator.This means that the confidence level represents the theoretical long-run frequency (i.e., the proportion) of confidence intervals that contain the true value of the unknown population parameter. In other words, 90% of confidence intervals computed at the 90% confidence level contain the parameter, 95% of confidence intervals computed at the 95% confidence level contain the parameter, 99% of confidence intervals computed at the 99% confidence level contain the parameter, etc.The confidence level is designated before examining the data. Most commonly, a 95% confidence level is used. However, other confidence levels, such as 90% or 99%, are sometimes used.
Factors affecting the width of the confidence interval include the size of the sample, the confidence level, and the variability in the sample. A larger sample will tend to produce a better estimate of the population parameter, when all other factors are equal. A higher confidence level will tend to produce a broader confidence interval.
Many confidence intervals are of the form



(
t
−
c

σ

T


,
t
+
c

σ

T


)


{\displaystyle (t-c\sigma _{T},t+c\sigma _{T})}
, where



t


{\displaystyle t}
is the realization of the dataset, c is a constant and




σ

T




{\displaystyle \sigma _{T}}
is the standard deviation of the dataset.Another way to express the form of confidence interval is a set of two parameters: (point estimate – error bound, point estimate + error bound), or symbolically expressed as (–EBM, +EBM), where (point estimate) serves as an estimate for m (the population mean) and EBM is the error bound for a population mean.The margin of error (EBM) depends on the confidence level.A rigorous general definition:
Suppose a dataset




x

1


,
…
,

x

n




{\displaystyle x_{1},\ldots ,x_{n}}
is given, modeled as realization of random variables




X

1


,
…
,

X

n




{\displaystyle X_{1},\ldots ,X_{n}}
. Let



θ


{\displaystyle \theta }
be the parameter of interest, and



γ


{\displaystyle \gamma }
a number between 0 and 1. If there exist sample statistics




L

n


=
g
(

X

1


,
…
,

X

n


)


{\displaystyle L_{n}=g(X_{1},\ldots ,X_{n})}
and




U

n


=
h
(

X

1


,
…
,

X

n


)


{\displaystyle U_{n}=h(X_{1},\ldots ,X_{n})}
such that:




P
(

L

n


<
θ
<

U

n


)
=
γ


{\displaystyle P(L_{n}<\theta <U_{n})=\gamma }
for every value of



θ


{\displaystyle \theta }

then



(

l

n


,

u

n


)


{\displaystyle (l_{n},u_{n})}
, where




l

n


=
g
(

x

1


,
…
,

x

n


)


{\displaystyle l_{n}=g(x_{1},\ldots ,x_{n})}
and




u

n


=
h
(

x

1


,
…
,

x

n


)


{\displaystyle u_{n}=h(x_{1},\ldots ,x_{n})}
, is called a



γ
×
100
%


{\displaystyle \gamma \times 100\%}
confidence interval for



θ


{\displaystyle \theta }
. The number



γ


{\displaystyle \gamma }
is called the confidence level.

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