Statistical Analysis for Education and Psychology Researchers

ard

error

propor

tions

—6.6

where P 1 and P 2 are the observed sample proportions for the two independent samples of
size n 1 and n 2. The sample proportion is the number of counts in the relevant category
divided by the sample size.

Confidence Intervals

Consider the first hypothesis, the difference in the proportions of respondents who
endorsed the category very much/much to the two questions. Advice on materials
(observed proportion P 1 =96/110), and INSET for school staff (observed proportion P 2 =
72/108).
The 95 per cent CI for the population difference in the two proportions (π 1 −π 2 ) is:
(P 1 −P 2 )−(Z×SD) to (P 1 −P 2 )+(Z×SD)^

where Z is the upper value from a standard normal distribution for the selected
100(1−α/2). For example, for a 95 per cent CI Z=1.96. Unlike the t-distribution this
standard normal critical value does not depend on the sample size.
In this example the standard error of the observed difference is:

The 95 per cent CI is
(P 1 −P 2 )−(Z×SD) to (P 1 −P 2 )+(Z×SD)
(0.873−0.667)−(1.96×0.055) to (0.873−0.667)+(1.96×0.055)
= (0.10 to 0.31).

Interpretation

The 95 per cent CI does not include the value zero and we can therefore conclude with 95
per cent certainty that the population proportions are significantly different, p<0.05. As
the investigators concluded, advice on materials for children with learning difficulties
was judged by primary school headteachers to be of greater importance than in-service
training for school staff.

Computer Analysis

Once again this analysis is accomplished with ease with a few lines of SAS code. A
complete SAS programme for the proportions test is shown in Figure 7, Appendix A3. In

Inferences involving binomial and nominal count data 187