  There are many sources of error in counting votes: the apparent winner might
not be the rightful winner. Hand tallies of the votes in a random sample of
precincts can be used to test the hypothesis that a full manual recount would
find a different outcome. This paper develops a conservative sequential test
based on the vote-counting errors found in a hand tally of a simple or
stratified random sample of precincts. The procedure includes a natural
escalation: If the hypothesis that the apparent outcome is incorrect is not
rejected at stage $s$, more precincts are audited. Eventually, either the
hypothesis is rejected--and the apparent outcome is confirmed--or all precincts
have been audited and the true outcome is known. The test uses a priori bounds
on the overstatement of the margin that could result from error in each
precinct. Such bounds can be derived from the reported counts in each precinct
and upper bounds on the number of votes cast in each precinct. The test allows
errors in different precincts to be treated differently to reflect voting
technology or precinct sizes. It is not optimal, but it is conservative: the
chance of erroneously confirming the outcome of a contest if a full manual
recount would show a different outcome is no larger than the nominal
significance level. The approach also gives a conservative $P$-value for the
hypothesis that a full manual recount would find a different outcome, given the
errors found in a fixed size sample. This is illustrated with two contests from
November, 2006: the U.S. Senate race in Minnesota and a school board race for
the Sausalito Marin City School District in California, a small contest in
which voters could vote for up to three candidates.
