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Prediction intervals are commonly used as definitions of reference ranges, such as reference ranges for blood tests to give an idea of whether a blood test is normal or not. For this purpose, the most commonly used prediction interval is the 95% prediction interval, and a reference range based on it can be called a standard reference range.
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Bootstrapping assigns measures of accuracy (bias, variance, confidence intervals, prediction error, etc.) to sample estimates. [2] [3] This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. [1]
Confidence bands can be constructed around estimates of the empirical distribution function.Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the Kolmogorov-Smirnov test, or by using non-parametric likelihood methods.
For an approximately normal data set, the values within one standard deviation of the mean account for about 68% of the set; while within two standard deviations account for about 95%; and within three standard deviations account for about 99.7%. Shown percentages are rounded theoretical probabilities intended only to approximate the empirical ...
Utilizing the likelihood-based method, confidence intervals can be found for exponential, Weibull, and lognormal means. Additionally, likelihood-based approaches can give confidence intervals for the standard deviation. It is also possible to create a prediction interval by combining the likelihood function and the future random variable. [3]
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
We can therefore use this quotient to find a confidence interval for μ. This t-statistic can be interpreted as "the number of standard errors away from the regression line." This t-statistic can be interpreted as "the number of standard errors away from the regression line."