Introduction to Probability and Statistics for Engineers and Scientists

516 Chapter 12:Nonparametric Hypothesis Tests

This hypothesis can easily be tested by noting that each of the observations will,
independently, be less thanm 0 with probabilityF(m 0 ). Hence, if we let

Ii=

{

1ifXi<m 0

0ifXi≥m 0

thenI 1 ,...,Inare independent Bernoulli random variables with parameterp=F(m 0 );
and so the null hypothesis is equivalent to stating that this Bernoulli parameter is equal to
1
2. Now, ifvis the observed value of

∑n
i= 1 Ii— that is, ifvis the number of data values
less thanm 0 — then it follows from the results of Section 8.6 that thep-value of the test
that this Bernoulli parameter is equal to^12 is

p-value=2 min(P{Bin(n, 1/2)≤v},P{Bin(n, 1/2)≥v}) (12.2.1)

where Bin(n,p) is a binomial random variable with parametersnandp.
However,

P{Bin(n,p)≥v}=P{n−Bin(n,p)≤n−v}

=P{Bin(n,1−p)≤n−v} (why?)

and so we see from Equation 12.2.1 that thep-value is given by

p-value=2 min(P{Bin(n, 1/2)≤v},P{Bin(n, 1/2)≤n−v}) (12.2.2)

=







2 P{Bin(n, 1/2)≤v} ifv≤

n

2

2 P{Bin(n, 1/2)≤n−v} ifv≥

n

2

Since the value ofv=

∑n
i= 1 Iidepends on the signs of the termsXi−m^0 , the foregoing
is called thesign test.

EXAMPLE 12.2a If a sample of size 200 contains 120 values that are less thanm 0 and 80
values that are greater, what is thep-value of the test of the hypothesis that the median is
equal tom 0?

SOLUTION From Equation 12.2.2, thep-value is equal to twice the probability that
binomial random variable with parameters 200,^12 is less than or equal to 80.
The text disk shows that

P{Bin(200, .5)≤ 80 }=.00284

Therefore, thep-value is .00568, and so the null hypothesis would be rejected at even the
1 percent level of significance. ■