Confidence Interval
المؤلف:
Kenney, J. F. and Keeping, E. S.
المصدر:
"Confidence Limits for the Binomial Parameter" and "Confidence Interval Charts." §11.4 and 11.5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand
الجزء والصفحة:
...
11-2-2021
1821
Confidence Interval
A confidence interval is an interval in which a measurement or trial falls corresponding to a given probability. Usually, the confidence interval of interest is symmetrically placed around the mean, so a 50% confidence interval for a symmetric probability density function would be the interval ![[-a,a]](https://mathworld.wolfram.com/images/equations/ConfidenceInterval/Inline1.gif)
such that
 |
(1)
|
For a normal distribution, the probability that a measurement falls within 
standard deviations (
) of the mean 
(i.e., within the interval ![[mu-nsigma,mu+nsigma]](https://mathworld.wolfram.com/images/equations/ConfidenceInterval/Inline5.gif)
) is given by
Now let 
, so 
. Then
where 
is the so-called erf function. The following table summarizes the probabilities 
that measurements from a normal distribution fall within ![[mu-x_n,mu+x_n]](https://mathworld.wolfram.com/images/equations/ConfidenceInterval/Inline25.gif)
for 
with small values of 
.
 |
 |
 |
0.6826895 |
 |
0.9544997 |
 |
0.9973002 |
 |
0.9999366 |
 |
0.9999994 |
Conversely, to find the probability-
confidence interval centered about the mean for a normal distribution in units of 
, solve equation (5) for 
to obtain
 |
(7)
|
where 
is the inverse erf function. The following table then gives the values of 
such that ![[mu-x_P,mu+x_P]](https://mathworld.wolfram.com/images/equations/ConfidenceInterval/Inline40.gif)
is the probability-
confidence interval for a few representative values of 
. These values can be returned by NormalCI[0, 1, ConfidenceLevel -> P] in the Wolfram Language package HypothesisTesting` .
 |
 |
| 0.800 |
 |
| 0.900 |
 |
| 0.950 |
 |
| 0.990 |
 |
| 0.995 |
 |
| 0.999 |
 |
REFERENCES:
Kenney, J. F. and Keeping, E. S. "Confidence Limits for the Binomial Parameter" and "Confidence Interval Charts." §11.4 and 11.5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 167-169, 1962.
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