Methods and formulas for 1 Proportion

Select the method of your choice.

Confidence interval (CI) for the exact method

Note

The interval (PL, PU) is an approximate 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

TermDescription
v1 2x
v2 2(nx + 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

TermDescription
v1 2(x + 1)
v2 2(nx)
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

TermDescription
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: ppo and po = 1/2, p-value = P{ X < y or X > ny | p = po}

Notation

TermDescription
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

TermDescription
c critical value chosen to obtain the desired significance level, α
x observed number of successes
y min {x, nx}
n number of trials

Test statistic for the normal approximation

Formula

Notation

TermDescription
observed probability, x/n
x observed number of events in n trials
n number of trials
p0 hypothesized probability
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