TESTING FOR EQUALITY OF MEANS: PAIRED DATA 107Table 8.3
and Their 10 SiblingsAge at First Word in Months for 10 Children with Cyanotic Heart DiseasePair Number Cyanotic XI Sibling Xz Difference d
1 2 3 4 5 6 7 8 91011.8
20.8
14.5
9.5
13.5
22.6
11.1
14.9
16.5
16.59.8 2.0
16.5 4.3
14.5 0.0
15.2 -5.7
11.8 1.7
12.2 10.4
15.2 -4.1
15.6 -0.7
17.2 -0.7
10.5 6.0- d = 1.32
sz = 22.488
In most statistical programs, one variable is used to list the hemoglobin levels
for both groups. A second variable, often called a grouping or class variable, tells
the computer program which group the observation comes from. In the hemoglobin
example, the class or grouping variable could be assigned numerical values of 1 or
2 to indicate whether the child was cyanotic or acyanotic. Alternatively, the data for
the two groups could be placed in two columns.
8.3 TESTING FOR EQUALITY OF MEANS: PAIRED DATA
Often in comparing two populations, the sample data occur in pairs. Researchers
studying congenital heart disease who wish to compare the development of cyanotic
children with normal children might gather data on age at first word for two inde-
pendent samples: one of cyanotic children and the other of well children. Then, we
could test the null hypothesis that the two population means are equal as in Section
8.2. Instead, however, available data (see Table 8.3) might consist of age at first word
for 10 pairs of children, with each pair consisting of a cyanotic child and a normal
sibling from the same family. Because children in the same family tend to be alike in
many respects, the data cannot be considered as two independent samples and should
be treated as a sample of pairs.
Instead of analyzing the ages directly, the 10 differences are formed as in Table 8.3
and are then treated as a single sample from a population of differences. If d represents
a difference and pd the mean of the population of differences, then if the cyanotic
children and siblings learn to talk at the same time, on the average, pd = 0.
Either a one- or a two-sided test can be performed. If the question to be answered
by the test is whether cyanotic children learn to talk later than siblings, a one-sided
test is appropriate. Rephrased in terms of ,ud, the question becomes: Is pd > O? A