# confidence interval & prediction interval & tolerance interval

confidence interval 是指估计的总体参数的可信度区间, 如总体均值,总体方差等;
prediction interval是指对于单个点估计的可信度区间.

http://www.cee.vt.edu/ewr/environmental/teach/smprimer/intervals/interval.html

#### Confidence Intervals

 Typically, since confidence intervals are based upon sample standard deviations, confidence interval calculations require sample sizes of four or more, as recommended by the EPA (EPA/530-R-93-003). Fewer data points result in wider confidence intervals, thus, larger sample sizes are preferred since a narrow interval is more useful. Remember, confidence intervals only apply to parameters, and not individual measurements. Thus, confidence intervals are only useful in estimating what the population parameter, such as the mean, should be; but it does not tell us anything about what any of the individual values in the population range from.

#### Tolerance Intervals

 Tolerance intervals are more applicable in areas such as compliance monitoring, because they tell us what the individual values should be. If the upper limit of a tolerance interval which is calculated from a sample set is higher than the set standard, then there is a high probability (1-gamma) that more than (alpha) percent of the measurements are above the standard, and thus, that the sight is in violation. As few as three data points can be used to generate a tolerance interval, but the EPA recommends having at lest eight points for the interval to have any usefulness (EPA/530-R-93-003).

#### Prediction Intervals

 As the name suggests, the prediction interval is useful in determining what future values should be, based upon present or past data. Prediction intervals are especially powerful because they can predict what a future compliance point should be less than before it is even collected, as opposed to having to wait until the data is collected in order to determine the tolerance interval and then comparing to standards. Another adantage is that as few as one future sample (k=1) can be used in determining the prediction interval, rather than a sample size of 8 or more for confidence or tolerance intervals. Thus, in areas such as groundwater monitoring, where a long period of time must pass, and few data points can be collected, prediction intervals are especially useful.

http://blog.sciencenet.cn/blog-54276-288414.html

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