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For Empirical Rule, 95% of the observed data will occur within the second standard deviation, but what is the difference between this and the 95/95 2-sided tolerance interval ? It includes 95% of population with 95% confidence. So, is the difference here the assigned confidence level ? And for Empirical Rule, does it tell us the percentage of the observed data only but not how confidence it is ?

These are basic concepts but I still confuse for it. Thanks

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  • $\begingroup$ Confidence interval: 95% of the time, the interval covers the parameter of interest of the distribution. Prediction interval: 95% of the time, the interval covers a new sample observation from the distribution. Tolerance interval: 95% of the time, the interval covers at least 95% of the distribution. As the original sample size increases, the confidence interval tends to shrink towards a point, the prediction interval tends to shrink towards a fixed interval based on the originally unknown distribution and the tolerance interval tends to shrink towards the prediction interval. $\endgroup$ Commented Jun 8, 2021 at 9:00
  • $\begingroup$ So, the Empirical Rule is actually the Prediction interval ? $\endgroup$ Commented Jun 8, 2021 at 14:00
  • $\begingroup$ The prediction interval needs to combine the uncertainty in the parameters with the uncertainty in the distribution. If the distribution is not normal then the empirical rule may never be justified and if the sample size is small then it may not be justified even for a normal distribution $\endgroup$ Commented Jun 8, 2021 at 14:14

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