The interval (PL, PU) is a 100(1 – α)% confidence interval of p.
When x = 0 or x = n, Minitab calculates only a one-sided confidence interval.
Lower limit
Formula
Notation
Term
Description
v1
2x
v2
2(n – x + 1)
x
number of events
n
number of trials
F
lower α/2 point of F distribution with v1 and v2 degrees of freedom
Upper limit
Formula
Notation
Term
Description
v1
2(x + 1)
v2
2(n – x)
x
number of events
n
number of trials
F
upper α/2 point of F distribution with v1 and v2 degrees of freedom
Confidence interval (CI) for the normal approximation
Formula
Notation
Term
Description
observed probability, = x / n
x
observed number of events in n trials
n
number of trials
zα/2
inverse cumulative probability of the standard normal distribution at 1–α/2
α
1 – confidence level/100
Exact test
Formula
The sample (X) comes from binomial distribution with parameters n and p.
H1: p > po,
p-value = P{ X ≥ x | p = po}
H1: p < po,
p-value = P{ X ≤ x | p = po}
H1: p ≠ po and po = 1/2, p-value = P{ X < y or X > n – y | p = po}
Notation
Term
Description
n
number of trials
p
probability of success
x
observed number of successes
y
min {x, n – x}
Likelihood ratio test
Formula
Minitab uses the likelihood ratio test for:
H1: p ≠ po and po ≠ 1/2
The likelihood function is defined as:
such that
LR (x) ≥ c
Minitab evaluates the likelihood ratio for all possible values of X = (0, 1,…, n) and sums the probabilities for all values for which the LR (y) ≥ LR (x).
p-value = Σ P{X = y | p = po}
Notation
Term
Description
c
critical value chosen to obtain the desired significance level, α
= x / n 
