Introduction to Probability and Statistics for Engineers and Scientists

12.3The Signed Rank Test 519

wherepis the probability that a randomly chosen member of the population has an annual
income of less than $90,000. Therefore, thep-value is

p-value=P{Bin(80, 1/2)≤ 28 }=.0048

and so the null hypothesis that the median income is less than or equal to $90,000 is
rejected. ■

A test of the one-sided null hypothesis that the median is at leastm 0 is obtained
similarly. If a random sample of sizenis chosen, andvof the resulting values are less
thanm 0 , then the resultingp-value is

p-value=P{Bin(n, 1/2)≥v}

12.3 The Signed Rank Test

The sign test can be employed to test the hypothesis that the median of a continuous
distributionFis equal to a specified valuem 0. However, in many applications one is really
interested in testing not only that the median is equal tom 0 but that the distribution is
symmetric aboutm 0. That is, ifXhas distribution functionF, then one is often interested
in testing the hypothesisH 0 : P{X <m 0 −a}=P{X > m 0 +a}for alla > 0
(see Figure 12.2). Whereas the sign test could still be employed to test the foregoing
hypothesis, it suffers in that it compares only the number of data values that are less than
m 0 with the number that are greater thanm 0 and does not take into account whether or
not one of these sets tends to be further away fromm 0 than the other. A nonparametric test
that does take this into account is the so-calledsigned ranktest. It is described as follows.
LetYi = Xi−m 0 ,i = 1,...,n and rank (that is, order) the absolute values
|Y 1 |,|Y 2 |,...,|Yn|, Set, forj=1,...,n.

Ij=







1 if thejth smallest value comes from a data value that is smaller

thanm 0

0 otherwise

Now, whereas

∑n
j= 1 Ijrepresents the test statistic for the sign test, the signed rank test
uses the statisticT=

∑n
j= 1 jIj. That is, like the sign test it considers those data values
that are less thanm 0 , but rather than giving equal weight to each such value it gives larger
weights to those data values that are farthest away fromm 0.

EXAMPLE 12.3a Ifn=4,m 0 =2, and the data values areX 1 =4.2,X 2 =1.8,X 3 =5.3,
X 4 =1.7, then the rankings of|Xi− 2 |are .2, .3, 2.2, 3.3. Since the first of these
values — namely, .2 — comes from the data pointX 2 , which is less than 2, it follows that