On Estimating the Size and Confidence of a Statistical Audit

**Working Paper No.**: 54

**Date Published**: 2008-11-30

#### Author(s):

Javed A. Aslam, Northeastern University

Raluca A. Popa, Massachusetts Institute of Technology

Ronald L. Rivest, Massachusetts Institute of Technology

#### Abstract:

We consider the problem of statistical sampling
for auditing elections, and we develop a remarkably
simple and easily-calculated upper bound
for the sample size necessary for determining
with probability at least c whether a given set
of n objects contains b or more “bad” objects.
While the size of the optimal sample drawn without
replacement can be determined with a computer
program, our goal is to derive a highly accurate
and simple formula that can be used by
election officials equipped with only a simple calculator.
We actually develop several formulae,
but the one we recommend for use in practice is:
U3(n, b, c)
=
ln −
(b − 1)
2
·
1 − (1 − c)
1/bm
=
ln −
(b − 1)
2
·
1 − exp(ln(1 − c)/b)
m
As a practical matter, this formula is essentially
exact: we prove that it is never too small, and
empirical testing for many representative values
of n ≤ 10, 000, and b ≤ n/2, and c ≤ 0.99 never
finds it more than one too large. Theoretically,
we show that for all n and b this formula never
exceeds the optimal sample size by more than 3
for c ≤ 0.9975, and by more than (− ln(1−c))/2
for general c.