Statistical Analysis for Education and Psychology Researchers

previous year. If we plan to select a simple random sample of 500 secondary schools,
what will be the expected mean and standard deviation of the number of schools with
unauthorized absence rates >1%?
An appropriate statistical model is the binomial probability model. Each school has a
probability of 0.5 and the binomial population from which the sample is selected is
B(500,0.5). The mean number of schools is evaluated using equation 4.2 for the mean of
a binomial variable.
μx=nπ=(500).(0.5)=250^

We would expect to find 250 schools in our sample with unauthorized absence rates

1%, and using equation 4.3, the standard deviation would be:

Sample Proportions and Percentages

Proportions and percentages are common in education and psychological research and the
sampling distributions of these statistics allow researchers to make statistical inferences
about proportions and percentages in the population.

Example 4.4: Sample Proportions and the Binomial Distribution

Consider pupil performance on interpreting graphs. Swatton and Taylor (1994) report that
30 per cent (total n=60) of age 11 pupils cannot correctly interpret graphs in which minor
grid lines represent values other than 1 or 10.
Should we wish to estimate the proportion of age 11 pupils in the population who can
not correctly interpret graphs with complex scales, that is estimate the proportion of
‘successes’ in the population, then we can use the binomial probability distribution
provided we make a minor change to the data. In this example, do not confuse the term
‘success’ which relates to ‘successes’ & ‘failures’ in a binomial sample, with the idea of
success meaning to be able to interpret graphs with minor grid lines.
A sample proportion does not have a binomial distribution because it is not a count.
To estimate π, the proportion of ‘successes’ in the population, we simply restate the
reported sample percentage as a count and use the sample proportion (P =count/n) as an
estimator of π. The sample proportion, P is an unbiased estimator of the population
proportion, π. More precisely, the mean of a sample proportion, μp of ‘successes’, say
those children who can not interpret graphs, equals the population proportion of
‘successes’, π.

The population proportion, π, who can not correctly interpret graphs with complex scales
is estimated by the observed proportion, P=18/60=0.3. If the 60 pupils in the study are a
simple random sample of age 11 pupils, then on average, we would expect 30 per cent of
the population not to be able to interpret complex scales on graphs.

How confident should we be with the precision of this estimate?

Probability and inference 99