96 TESTS OF HYPOTHESES ON POPULATION MEANS
Table 8.1 Age of Walking for 18 Acyanotic ChildrenChild Age (months) Child Age (months)
15.0
11.0
14.2
10.0
12.0
14.2
14.5
13.8
12.810
11
12
13
14
15
16
17
1810.2
14.0
14.8
14.2
16.0
13.5
9.2
15.0
16.5- X = 13.38 s = 2.10
8.1 TESTS OF HYPOTHESES FOR A SINGLE MEAN
In this section we present tests that are used when only one mean is being tested:
one test for when the population standard deviation u is known and one where it is
unknown. We also discuss the usual procedures that are followed in tests concerning
means.
8.1.1 Test for a Single Mean When n Is Known
In a study of children with congenital heart disease, researchers gathered data on
age in months of walking from 18 children with acyanotic congenital heart disease;
the data are recorded in Table 8.1. From larger studies on normal children, the
researchers know that the mean age of walking for normal children is 12.0 months
and that the standard deviation is 1.75 months. We wish to decide whether or not,
on average, acyanotic children learn to walk at the same age as normal children. We
formulate a hypothesis that we propose to test. This proposed hypothesis is called a
null hypothesis. Our null hypothesis is as follows: The mean age of walking for a
population of acyanotic children is 12.0 months. The null hypothesis is equivalent
to saying that acyanotic children learn to walk at the same age on average as normal
children. On the basis of the data, we wish either to accept the null hypothesis or
to reject the null hypothesis. The null hypothesis should be decided upon prior to
analysis of the data.
The null hypothesis to be tested is often written in symbols, here HO : p = po.
HO denotes the null hypothesis, p denotes the mean of the population being studied,
and po denotes the hypothesized mean. In the acyanotic children example, the null- hypothesis is HO : p = 12.0. At this point, the student might suggest that we compute
X from the sample and see whether or not it is equal to 12.0. But because of sampling
variation, even if the average age of walking for the population of acyanotic children
is 12.0 months, this particular sample of 18 children may have 7 either higher or