Technically, the null hypothesis is tested using a pooled estimate of the standard error
of the difference in proportions because the null hypothesis actually states that the
population proportions are equal, π 1 =π 2. When a pooled estimate is used, the total
population proportion is estimated using the information contained in the two samples. In
fact a weighted mean proportion is used (the overall proportion of successes in the two
samples). This is the procedure most often presented in introductory statistical texts. For
reasons already mentioned, it is this author’s belief that whenever possible it is preferable
to give a confidence interval for any difference accordingly, the test procedure presented
here is to evaluate a confidence interval for the difference in proportions. This leads to a
test of essentially the same null hypothesis of no difference between proportions but
additionally provides a zone of confidence for any population difference in proportions.
The reader should note that the main difference between the confidence interval approach
and a direct test of the null hypothesis is the procedure used for estimating the standard
error.
Test Assumptions- Observations are sampled at random from a specified binary population. The population
can be treated as binary with respect to a continuous variable provided that values for
the statistical variable can be assigned to two mutually exclusive categories. - Each observation is independent (does not effect the value of any other observations
sampled). - This test is based on a normal approximation to the binomial distribution (normal
variate Z is used). The test procedure should not therefore be used when sample sizes
are small, say <25, or when the proportions are outside the range 0.1 to 0.9.
Examples from the LiteratureIn a study of social-cognitive modes of play Roopnarine et al. (1992) observed playmate
selection in different classroom structures (same age/mixed age classrooms). The study
included analysis of the proportions of child-initiated play activities with same sex peers
in same age and mixed-age classrooms. The investigators performed tests of differences
between the proportions of initiated play activities in the two different classroom
organizations (two independent samples of observations). The null hypothesis here is that
the proportions of initiated play activities with same sex peers would be equal in same
age and mixed-age classrooms. The alternative two-tailed hypothesis was that the
proportions would be different in the two types of classroom organization. The
investigators reported two Z-scores of 2.29, p<0.05 (boys) and 2.22, p<0.05 (girls) and
concluded that both 4-year-old-boys and 4-year-old-girls in same age classrooms were
more likely to initiate play with same sex peers than were their age equivalent
counterparts in mixed-age classrooms.
In a second example where percentage differences rather than proportions were
analysed MacKay and Boyle (1994) report on an investigation of primary headteachers’
expectations of their local education authorities educational psychologists. Specifically
the investigators looked at the importance placed by headteachers on provision of advice
on materials for pupils with learning difficulties.
Inferences involving binomial and nominal count data 185