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688 CHAPTER 13 NONPARAMETRIC AND DISTRIBUTION-FREE STATISTICS
TABLE 13.3.2 Scores Above () and Below () the Hypothesized Median Based
on Data of Example 13.3.1
Girl 1 2 3 4 5 6 7 8 9 10
Score relative to - 0
hypothesized
median
+
+
+
+
+
+
+
+
approximately equal. This line of reasoning suggests an alternative way in
which we could have stated the null hypothesis, namely, that the probability
of a plus is equal to the probability of a minus, and these probabilities are
equal to .5. Stated symbolically, the hypothesis would be
H 0 : P1+2 = P1-2 = .5
In other words, we would expect about the same number of plus signs as
minus signs in Table 13.3.2 when H 0 is true. A look at Table 13.3.2 reveals
a preponderance of pluses; specifically, we observe eight pluses, one minus,
and one zero, which was assigned to the score that fell exactly on the median.
The usual procedure for handling zeros is to eliminate them from the analysis
and reduce n, the sample size, accordingly. If we follow this procedure,
our problem reduces to one consisting of nine observations of which eight
are plus and one is minus.
Since the number of pluses and minuses is not the same, we wonder if
the distribution of signs is sufficiently disproportionate to cast doubt on our
hypothesis. Stated another way, we wonder if this small a number of minuses
could have come about by chance alone when the null hypothesis is true, or if
the number is so small that something other than chance (that is, a false null
hypothesis) is responsible for the results.
Based on what we learned in Chapter 4, it seems reasonable to conclude
that the observations in Table 13.3.2 constitute a set of n independent
random variables from the Bernoulli population with parameter p. If we let
k = the test statistic, the sampling distribution of k is the binomial probability
distribution with parameter p = .5 if the null hypothesis is true.
6. Decision rule. The decision rule depends on the alternative hypothesis.
For H A : P1+2 7 P1-2, reject H 0 if, when H 0 is true, the probability
of observing k or fewer minus signs is less than or equal to a.
For H A : P1+2 6 P1-2, reject H 0 if the probability of observing,
when H 0 is true, k or fewer plus signs is equal to or less than a.
For H A : P1+2 Z P1-2, reject H 0 if (given that H 0 is true) the probability
of obtaining a value of k as extreme as or more extreme than
was actually computed is equal to or less than a>2.
For this example the decision rule is: Reject H 0 if the p value for the
computed test statistic is less than or equal to .05.