For the distribution on the left of Figure 5.5, the stimulus is set at a speed that just barely
allows the participant to respond at better than chance levels, with a probability of .60 of be-
ing correct on any given trial. The distribution in the middle represents the results expected
from a judge who has a probability of only .30 of being correct on each trial. The distribu-
tion on the right represents the behavior of a judge with a nearly unerring ability to choose
the wrong stimulus. On each trial, this judge had a probability of only .05 of being correct.
From these three distributions, you can see that, for a given number of trials, as pand qde-
part more and more from .50, the distributions become more and more skewed although the
mean and standard deviation are still Npand respectively. Moreover, it is important
to point out (although it is not shown in Figure 5.5, in which Nis always 10) that as the num-
ber of trials increases, the distribution approaches normal, regardless of the values of pand q.
As a rule of thumb, as long as both Npand Nqare greater than about 5, the distribution is
close enough to normal that our estimates won’t be far in error if we treat it as normal.
Figure 5.6 shows the binomial distribution when p 5 .70 and there are 25 trials.
5.9 Using the Binomial Distribution to Test Hypotheses
Many of the situations for which the binomial distribution is useful in testing hypotheses
are handled equally well by the chi-square test, discussed in Chapter 6. For that reason, this
discussion will be limited to those cases for which the binomial distribution is uniquely
useful.
In the previous sections, we dealt with the situation in which a person was judging very
brief stimuli, and we saw how to calculate the distribution of possible outcomes and their
probabilities over N 5 10 trials. Now suppose we turn the question around and ask whether
the available data from a set of presentation trials can be taken as evidence that our judge
really can identify presented characters at better than chance levels.
For example, suppose we had our judge view eight stimuli, and the judge has been cor-
rect on seven out of eight trials. Do these data indicate that she is operating at a better than
1 Npq,
Section 5.9 Using the Binomial Distribution to Test Hypotheses 131
Probability
0.15
0.10
0.05
0.00
51015
Number of successes
20 25
Figure 5.6 Binomial distribution with p 5 .70 and n 525
