  The high fraction of published results that turn out to be incorrect is a
major concern of today's science. This paper contributes to the understanding
of this problem in two independent directions. First, Johnson's recent claim
that hypothesis testing with a significance level of 0.05 can alone lead to an
unacceptably large proportion of false positives among all results is shown to
be unfounded. Second, a way to quantify the effect of "positivity bias" (the
tendency to consider only positive results as worthwhile) is introduced. We
estimate the proportion of false positives among positive results in terms of
the significance level used and the positivity ratio. The latter quantity is
the fraction of positive results over all results, be they positive or not,
published or not. In particular, if one uses a significance level of 0.05, and
produces 4 (possibly unpublished) negative results for every positive result,
then the proportion of false positives among positive results can climb to a
high 21%.
