Basic Statistics

192 NONPARAMETRIC STATISTICS

The null hypothesis that the health plan had no effect on the number of days off work,
7r = 1/2 = .5, can be tested by computing

= 5(.03125) = .15625

and the probability that X = 0 is

The cumulative probability that X = 1 or less is .03125 + .15625 = .1875, so for
a one-sided test with Q = .05 we would not reject the null hypothesis. For a two-
sided test with Q = .025 on both sides, we also would not be able to reject the
null hypothesis. Note that there is no need to perform these calculations, as the
probabilities can be obtained from binomial tables by looking at the column that has
p = .5 and cumulating the values found for the X values that you want. For example,
for T = .5 and n = 5 that we just calculated, tabled values for X = 0 are given
as .03125 and for n = 1 as .15625, so all that needs to be done is to sum these two
values to get .1875.
Binomial tables may be found in Conover [1999], van Belle et al. [2004], Daniel
[1978], and many other statistic books. Dixon and Massey [1983] in their Table A. 10a
give critical values for the sign test.
In some texts and software programs, the null hypothesis is stated in terms of the
median M as Ho : M = iUo. Note that the observations being tested are the number
of plus and minus signs of the n differences of X, and Y,. If the number of plus and
minus signs are equal, half are above zero and half below (see Section 5.1.2 for the
definition of a median). For a further explanation of the interpretation of this test, see
Conover [1999] or Gibbons [1993]. Minitab, SAS, SPSS, Stata, and StatXact will
perform the sign test.

13.2 THE WILCOXON SIGNED RANKS TEST

The Wilcoxon signed ranks test is used when the data in two samples are paired. It
is often used when the assumptions for the paired Student t test are not met. Note
that the sign test uses only information on the sign of the differences. The Wilcoxon
signed ranks test gives more weight to a pair with large differences between the two
paired observations. However, the Wilcoxon signed ranks test does require more
assumptions than the sign test.

13.2.1 Wilcoxon Signed Ranks Test for Large Samples

The data to be testedconsists of n paired observations (XI, Yl), (Xz,Yz),... , (Xn,Yn).
The numerical values of D, = U, - X, are then computed and the absolute differences
1 D, I = /X, - 1, which are always positive, are obtained.