For a sample of ten outcrops, five of which are above the K/T boundary
and five below, together with the constraint that four outcrops must have
thick-shelled molluscs and six must lack them, there are five possible out-
comes (Table 18.6). To obtain these, you start with the outcome expected
under the null hypothesis (c), choose one of the four cells (it does not matter
which) and add one to that cell. Next, adjust the values in the other three
cells so the marginal totals do not change. Continue with this procedure
until the number within the cell you have chosen cannot be increased any
further without affecting the marginal totals. Then go back to the expected
outcome and repeat the procedure by subtracting one from the same cell
until the number in it cannot decrease any further without affecting the
marginal totals (Table 18.6).
Third, the actual outcome is identified within the total set of possible
outcomes. For this example, it is case (e) in Table 18.6. The probability of
this outcome, together with any more extreme departures in the same
direction from the one expected under the null hypothesis (here there are
none more extreme than (e)) can be calculated from the probability of
getting this particular arrangement within the four cells by sampling a set
of ten outcrops, four of which contain thick-shelled molluscs and six of
which do not, with the outcrops sampled from above and below the K/T
boundary. This is similar to the example used to introduce hypothesis
testing in Chapter 6, where you had to imagine a sample of hornblende
vs. quartz grains in a beach sand. Here, however, a very small group is
sampled without replacement, so the initial probability of selecting an out-
crop with thick-shelled molluscs present is 4/10, but if one is drawn, the
probability of next drawing an outcrop with thick-shelled molluscs is now
3/9 (and 6/9 without). We deliberately have not given this calculation
because it is long and tedious, and most statistical packages do it as part
of the Fisher Exact Test.
The calculation gives the exact probability of getting the observed out-
come or a more extreme departure in the same direction from that expected
under the null hypothesis. This is a one-tailed probability, because the
outcomes in the opposite direction (e.g. on the left of (c) in Table 18.6)
have been ignored. For a two-tailed hypothesis you need to double the
probability. When the probability is less than 0.05, the outcome is consid-
ered statistically significant.
18.4 Bias when there is one degree of freedom 241