The following three questions were included in a questionnaire survey of primary
school headteachers (with 96 per cent return rate). Respondents’ replies as reported by
the investigators are shown:
To what extent do you feel that the Psychologist should be involved in the
following areas?(^) n=(100%) Very much/ much
involved
in
between
Little/ not
involved
% n % n % n
Q1 Advice on materials (for pupils with
learning difficulties)
110 87.3 96 11.8 13 0.9 1
Q2 INSET for school staff (in-service
education and training)
108 66.7 72 26.9 29 6.4 7
Q3 Primary/secondary school liaison 109 65.1 71 26.6 29 8.3 9
The investigators reported that provision of advice was judged to be of significantly
greater importance than either INSET or primary-secondary school liaison (p<0.05).
Worked Example
Data from the study of headteachers’ perceptions will be used to illustrate application of
the proportions test for detecting the significance of any difference between
percentages. As the investigators indicate that the response rate is 96 per cent, it is
reasonable to assume that the achieved sample is representative of headteachers in the
LEA(s) concerned. We also need to assume that the sample is random and that responses
to the three questions are independent, that is, a headteacher’s response to the third
question is not influenced by his or her response to the first or second questions. The
samples of responses are sufficiently large for the normal approximation to apply. The
investigators do not state precisely the comparisons made, but it is reasonable to assume
that the proportion of respondents endorsing the category very much/much involved in Q
1 is compared with proportions of responses in the same categories in Q 2 and 3. Thus
two null hypotheses are tested at the 5 per cent level:
H 0 : Q1 πvery much/much=Q2 πvery much/much
H 0 : Q1 πvery much/much=Q3 πvery much/much
It is also assumed that the alternative hypotheses are two-tailed tests.
The sample standard deviation of the difference in proportions, SD, that is the
standard error of the difference in proportions is used to estimate the unknown
population parameter, the standard deviation (or standard error) of the difference in
proportions, σD. That is SD (from sample) estimates σD in the population. The
computational formula for SD is:
of
differe
nce in
Stand
Statistical analysis for education and psychology researchers 186