Introduction to Probability and Statistics for Engineers and Scientists

520 Chapter 12:Nonparametric Hypothesis Tests

1.0

0.8

0.6

0.4

0.2

0.0 1

x

0 23456

y

FIGURE 12.2 A symmetric density: m= 3.

f(x)=

{

max{0, .4(x−3)+

√

.4} x≤ 3

max{0,−.4(x−3)+

√

.4} x> 3

I 1 =1. Similarly,I 2 =1, andI 3 andI 4 equal 0. Hence, the value of the test statistic is
T= 1 + 2 =3. ■

WhenH 0 is true, the mean and variance of the test statisticTare easily computed.
This is accomplished by noting that, since the distribution ofYj=Xj−m 0 is symmetric
about 0, for any given value of|Yj|— say,|Yj|=y— it is equally likely that eitherYj=y
orYj=−y. From this fact it can be seen that underH 0 ,I 1 ,...,Inwill be independent
random variables such that

P{Ij= 1 }=^12 =P{Ij= 0 }, j=1,...,n

Hence, we can conclude that underH 0 ,

E[T]=E





∑n

j= 1

jIj





=

∑n

j= 1

j

2

=

n(n+1)

4

(12.3.1)