Statistical Analysis for Education and Psychology Researchers

test would be to determine whether the average change (average of the differences
before-after scores) in scores is greater than would be expected due to chance fluctuations
alone.
The paired t-test is based on the same idea as the independent t-test, the test statistic is
a ratio of mean difference (predicted variability) to the standard error of the difference
(overall variability in scores). When the same subjects are used for both measurements
the standard error is smaller (a desirable research design feature) and consequently
smaller differences in means are likely to be detected. With fewer than five pairs of
scores the test is not very sensitive. Large differences in scores are needed to detect a
significant difference and this procedure should not be used when the population of
differences is non-normal.

Statistical Inference and Null Hypothesis

The sampling distribution of the difference scores (represented by D) is used as the basis
for statistical inference in the paired t-test. The mean of the population of difference
scores, μD, is zero, when the null hypothesis is true. We think of this as a one sample test
even though we are comparing two means because we have one population distribution
of difference scores. The null hypothesis can be written as H 0 :μD=μ 1 −μ 2 =0. There are
three possible alternative hypotheses:

1 μD≠0 a non directional test (two tailed), rejection region is |t|>t 1 −α/2.
2 μD<0 directional test (one tailed) rejection region t>t 1 −α or t<−t 1 −α.
3 μD>0 directional test (one tailed) rejection region t>t 1 −α or t<−t 1 −α.

Test Assumptions

The paired t-test should be considered when the population of interest consists of
difference scores from paired observations; this implies continuous measurement. The
following assumptions should be met:

• Paired differences are randomly selected from the population. This usually means that
  the sample is drawn at random.


• The population of difference scores is approximately normally distributed.


• Observations within a treatment condition are independent of each other.

Example from the Literature

Borzone de Manrique and Signorini (1994) compared two measures, spelling and
reading, within groups, using the t-test for paired observations. Scores analysed were
percentage correct to allow for comparisons across the spelling and reading tests which
had different numbers of items. Two paired t-tests were performed on separate groups of
pupils, a group of skilled readers, n=19, (score at the 75th percentile or better on a
standardized reading comprehension test) and a group of less skilled readers, n=20.
The authors reported a significant difference between spelling and reading in the less
skilled group, t(19)=5.24, p<0.001, but no difference was found in the skilled group,
t(18)=1.63, not significant. The authors concluded that the skilled readers perform

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