When a significant association between two variables is detected, it is sensible to
consider whether a third variable might explain this association. If this variable is in the
data set, a three-way frequency table could be produced stratifying on the third variable.
For example, if the variable ‘teaching experience’ (number of years as a classroom
teacher) was measured in the study by Cope, et al., (1992) this might have some
explanatory power for the apparent relationship between gender and increased
understanding. If a study used several different Chi-square tests, it would be advisable to
adjust the probability level associated with each statistical test (make it more
conservative) to take account of a significant result occurring simply by chance.
6.2 Binomial TestWhen to UseThe Binomial test is not widely used by educational researchers but is suitable when a
single random sample is selected from a binary population and each sampled observation
can be classified into one of two mutually exclusive categories. The sample proportion of
observations in one of the two categories is used to estimate the population proportion in
the same category. As this is a one-sample test, both proportions (or percentages) must
equal one (or 100 per cent).
The test is particularly useful when it is believed that the population proportion falling
into one of the two categories is 0.5. This is, in effect, a hypothesis of no difference in the
proportion of responses in the two categories, that is P 1 =P 2 =0.5, which would equal the
population proportion. If P 1 equals the proportion of observations in one category then
1 −P 1 (sometimes called Q) is the proportion of observations in the other category.
Often research designs involve comparison between matched groups on a binary
variable of interest. For example, following the introduction of student loans, an
investigator may be interested in whether students incur serious financial debts during
their time at college. The binary variable would be, in serious debt/not in serious debt. A
random sample of fourteen males might be matched with a random sample of fourteen
females, same age, same college and the proportion of males in debt compared with the
proportion of females who were in debt. The achieved sample might be:
(^) In serious debt at college Total
Male 10 14
Female 6 14
100 per cent 16 28
The null hypothesis is that the population proportions are equal, that is there is no
difference between the probability of males and the probability of females who incur
serious debt problems during college. If the null hypothesis were true we would expect to
find in the sample of 16 students who were in debt, eight males and eight females.
The binomial test is also useful for analyzing responses to multiple choice questions.
Given ten true/false multiple choice questions, a teacher may want to know how many
correct answers would be expected if a candidate were guessing at random. (Caution is
Inferences involving binomial and nominal count data 173