Statistical Analysis for Education and Psychology Researchers

required because candidates often do not guess at random; they may use partial
knowledge.)

Statistical Inference and Null Hypothesis

The parameters estimated are population proportions. The null hypothesis is often H 0 : π
(population proportion)=0.5. If the proportion of responses in category 1 is P and the
proportion in category 2 is Q, then there are three alternative hypotheses: i) H 1 : P>Q, ii)
H 1 : P<Q iii) H 1 : P≠Q. A one-tailed test is used when we predict in advance which of the
two categories should contain the smaller number of counts (i and ii above). If the
alternative hypothesis is simply that the counts in the two categories will differ (iii above)
then a two-sided test should be used. The sampling distribution used is the binomial
distribution, in this example B(n, π) would be Binomial (16, 0.5) see Chapter 4, section
4.4 for details.

Test Assumptions

The test assumptions are those of the binomial distribution which are as follows:

1 Observations are sampled at random from a binary population.
2 Each observation is independent (does not effect the value of any other observations
sampled)
3 The probability of any sample observation being classified into one of the two
categories is fixed for the population.
4 With small sample sizes such as n≤25, the exact binomial probability can be evaluated.
With larger sample sizes, especially when P is close to 0.5, the binomial
approximation to the normal distribution with a continuity correction (because the
normal distribution is continuous but the binomial distribution is discrete) can be used.
In this case the normal variate Z is used to evaluate the probability of the observed
outcome.

Example from the Literature

Blasingame and McManis (1977), in a study of retarded adults, investigated
developmental aspects of cognition, specifically transitivity of inequality (A>B, B>C,
therefore A>C), classification (subjects had to select a geometric figure to complete a
missing cell in a 2×2 matrix) and relative thinking (measured by the right-left test,
Elkind, 1961). Sequentiality of development among the three tasks was evaluated by
cross classification of subjects on pairs of tasks according to performance (achieve/not
achieve). The number of subjects performing discrepantly in each direction on two tasks
(achieve one task but not the other) was evaluated for statistical significance using the
binomial test. The authors give the following results:
Discrepant Performance between relative thinking and transivity tasks Number of retarded adults
transitivity achieved but relative thinking not achieved 4
transitivity not achieved but relative thinking achieved 13
100 per cent 17

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