# 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 $$m$$

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 