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Table 1 Confidence interval (CI), prediction interval (PI), and tolerance interval (TI) estimates for horn: \(\widehat{\pi }=x/n=2/16=0.125=12.5\%\) with confidence level \(\left(1-\alpha \right)\) = 0.95, classical Wilson (W) and Bayesian Jeffreys (J) for different contents \(P\), and different numbers of predicted future observations \(m\)

From: Teaching: confidence, prediction and tolerance intervals in scientific practice: a tutorial on binary variables

Type \(P\) \(m\) Length count scale Lower bound count
(\(CLB\))
Upper bound count (\(CUB\)) Lower bound (\(LB\))
\(\pi \)
Upper bound
(\(UB\))
\(\pi \)
Length
\(\pi \) scale
W-CIc    6a 0a 6a 0.034 0.360 0.326
J-CId    6a 0a 6a 0.026 0.344 0.318
J-PIe   50 19 0 19 0b 0.38b 0.38b
J-PIe   100 34 2 36 0.02b 0.36b 0.34b
W-TIf 0.8 50 22 0 22 0b 0.44b 0.44b
W-TIf 0.8 100 41 1 42 0.01b 0.42b 0.41b
W-TIf 0.9 50 24 0 24 0b 0.48b 0.48b
W-TIf 0.9 100 43 1 44 0.01b 0.44b 0.43b
J-TIg 0.8 50 22 0 22 0b 0.44b 0.44b
J-TIg 0.8 100 40 1 41 0.01b 0.41b 0.40b
J-TIg 0.9 50 23 0 23 0b 0.46b 0.46b
J-TIg 0.9 100 42 0 42 0b 0.42b 0.42b
  1. Original bounds are marked in bold
  2. a \(CLB=n*LB\) and \(CUB=n*UB\) lead to CI for the counts
  3. b \(LB=CLB/m\) and \(UB=UB/m\) lead to PI and TI for \(\pi \)
  4. cW-CI: classical Wilson confidence interval
  5. dJ-CI: Bayesian Jeffreys credible interval
  6. eJ-PI: Bayesian Jeffreys prediction interval
  7. fW-TI: classical Wilson tolerance interval
  8. gJ-TI: Bayesian Jeffreys tolerance interval