Confidence interval

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{{Ficontent}}
 
{{Ficontent}}
For an estimation, the confidence interval defines an upper and lower limit within which the true (population) value is expected to come to lie with a defined probability. This probability is frequently set to 95%, meaning that an error of <math>\alpha=5%</math> is accepted (other <math>\alpha</math> are also possible, of course).  
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For [[statistical estimations]], the confidence interval defines an upper and lower limit within which the true (population) value is expected to come to lie with a defined probability. This probability is frequently set to 95%, meaning that an error of <math>\alpha=5\%</math> is accepted (other <math>\alpha</math> are also possible, of course).  
  
 
In order to be able to build such a confidence interval, the distribution of the estimated values need to be known. It is known in sampling statistics, that the estimated mean follows a [[normal distribution]] if the sample is large (n>>30, say), and the t distribution with ν degrees of freedom when the sample is small (n<30, say).
 
In order to be able to build such a confidence interval, the distribution of the estimated values need to be known. It is known in sampling statistics, that the estimated mean follows a [[normal distribution]] if the sample is large (n>>30, say), and the t distribution with ν degrees of freedom when the sample is small (n<30, say).
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:<math>t_{\alpha v}*S_\bar{y}</math>
 
:<math>t_{\alpha v}*S_\bar{y}</math>
  
where <math>t</math> comes from the t-distribution and depends on [[sample size]] (df = degrees of freedom=n-1) and the error probability. Then,
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where <math>S_\bar y</math> is the [[standard error]] and <math>t</math> comes from the t-distribution and depends on [[sample size]] (df = degrees of freedom=n-1) and the error probability. Then,
  
:<math>P(\bar y -t*S_\bar{y}< \mu <\bar y +t*S_\bar{y})=95%</math>
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:<math>P(\bar y -t S_\bar{y}< \mu <\bar y +t S_\bar{y})=95\%</math>
  
 
As with the standard error of the mean, the width of the confidence interval (CI) can be given in absolute (in units of the mean value) or in relative terms (in %, relative to the estimated mean).   
 
As with the standard error of the mean, the width of the confidence interval (CI) can be given in absolute (in units of the mean value) or in relative terms (in %, relative to the estimated mean).   
  
If an estimation is accompanied by a precision statement, one must clearly say whether that is the standard error or half the width of the confidence interval!
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{{info
For larger sample sizes and α=5%, the t-value is , that is around 2 so that as a rule of thumb we may say that half the width of the confidence interval is given by twice the standard error: . For smaller sample sizes, the t-value will be larger.
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|message=Note!:
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|text=If an estimation is accompanied by a precision statement, one must clearly say whether that is the standard error or half the width of the confidence interval!}}
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For larger sample sizes and α=5%, the t-value is <math>t_{\alpha=0.05, v>30}=1.96</math>, that is around 2 so that as a rule of thumb we may say that half the width of the confidence interval is given by twice the standard error <math>\bar y \pm 2S_\bar y</math>. For smaller sample sizes, the t-value will be larger.
  
  

Latest revision as of 11:35, 20 December 2015

For statistical estimations, the confidence interval defines an upper and lower limit within which the true (population) value is expected to come to lie with a defined probability. This probability is frequently set to 95%, meaning that an error of \(\alpha=5\%\) is accepted (other \(\alpha\) are also possible, of course).

In order to be able to build such a confidence interval, the distribution of the estimated values need to be known. It is known in sampling statistics, that the estimated mean follows a normal distribution if the sample is large (n>>30, say), and the t distribution with ν degrees of freedom when the sample is small (n<30, say).

For a defined value \(\alpha\) for the error probability the width of the confidence interval for the estimated mean is given by

\[t_{\alpha v}*S_\bar{y}\]

where \(S_\bar y\) is the standard error and \(t\) comes from the t-distribution and depends on sample size (df = degrees of freedom=n-1) and the error probability. Then,

\[P(\bar y -t S_\bar{y}< \mu <\bar y +t S_\bar{y})=95\%\]

As with the standard error of the mean, the width of the confidence interval (CI) can be given in absolute (in units of the mean value) or in relative terms (in %, relative to the estimated mean).


info.png Note!:
If an estimation is accompanied by a precision statement, one must clearly say whether that is the standard error or half the width of the confidence interval!

For larger sample sizes and α=5%, the t-value is \(t_{\alpha=0.05, v>30}=1.96\), that is around 2 so that as a rule of thumb we may say that half the width of the confidence interval is given by twice the standard error \(\bar y \pm 2S_\bar y\). For smaller sample sizes, the t-value will be larger.

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