Introduction to Probability and Statistics for Engineers and Scientists

536 Chapter 12:Nonparametric Hypothesis Tests

and, similarly,

PH 0 {R≥r}≈ 1 −

(

r−μ

σ

)

Hence, for largenandm, thep-value of the runs test for randomness is approximately
given by

p-value≈2 min

{

(

r−μ

σ

)

,1−

(

r−μ

σ

)}

whereμandσare given by Equation 12.5.2 andris the observed number of runs.

EXAMPLE 12.5c Suppose that a sequence of sixty 1’s and sixty 0’s resulted in 75 runs. Since

μ=61 and σ=

√

3,540

119

=5.454

we see that the approximatep-value is

p-value≈2 min{ (2.567), 1− (2.567)}

= 2 ×(1−.9949)

=.0102

On the other hand, by running Program 12.5 we obtain that the exactp-value is

p-value=.0130

If the number of runs was equal to 70 rather than 75, then the approximatep-value would
be

p-value≈ 2 [ 1 − (1.650)]=.0990

as opposed to the exact value of

p-value=.1189 ■