Biostatistics

674 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS
Solution:
1. Data. See problem statement.
2. Assumptions. We assume that the measurements are taken on a
continuous variable.
3. Hypotheses.
H 0 : The population median is 5:
H A : The population median is not 5:
Let a ¼ :05.
4. Test statistic. The test statistic for the sign test is either the observed
number of plus signs or the observed number of minus signs. The nature
of the alternative hypothesis determines which of these test statistics
is appropriate. In a given test, any one of the following alternative
hypotheses is possible:
H A : PðþÞ > ð Þ one-sided alternative
H A : PðþÞ < ð Þ one-sided alternative
H A : PðþÞ 6¼ 1ð Þ two-sided alternative
If the alternative hypothesis is
H A : PðþÞ > Pð Þ
a sufficiently small number of minus signs causes rejection of H 0 . The
test statistic is the number of minus signs. Similarly, if the alternative
hypothesis is
H A : PðþÞ < Pð Þ
a sufficiently small number of plus signs causes rejection of H 0 . The test
statistic is the number of plus signs. If the alternative hypothesis is
H A : PðþÞ 6¼ Pð Þ
either a sufficiently small number of plus signs or a sufficiently small
number of minus signs causes rejection of the null hypothesis. We may
take as the test statistic the less frequently occurring sign.
5. Distribution of test statistic. As a first step in determining the nature of
the test statistic, let us examine the data in Table 13.3.1 to determine
which scores lie above and which ones lie below the hypothesized
median of 5. If we assign a plus sign to those scores that lie above the
hypothesized median and a minus to those that fall below, we have the
results shown in Table 13.3.2.
If the null hypothesis were true, that is, if the median were, in fact,
5, we would expect the numbers of scores falling above and below 5 to be