Introduction to Probability and Statistics for Engineers and Scientists

12.5The Runs Test for Randomness 535

are distinct, thenn=m=N/2.) DefineI 1 ,...,INby

Ij=

{

1ifXj≤s-med

0 otherwise

Now, if the original data constituted a random sample, then the number of runs in
I 1 ,...,INwould have a probability mass function given by Equation 12.5.1. Thus, it
follows that we can use the preceding runs test on the data valuesI 1 ,...,INto test that
the original data are random.

EXAMPLE 12.5b The lifetime of 19 successively produced storage batteries is as follows:

145 152 148 155 176 134 184 132 145 162 165

185 174 198 179 194 201 169 182

The sample median is the 10th smallest value — namely, 169. The data indicating whether
the successive values are less than or equal to or greater than 169 are as follows:

1111010111100000010

Hence, the number of runs is 8. To determine if this value is statistically significant, we
run Program 12.5 (withn=10,m=9) to obtain the result:

p-value=.357

Thus the hypothesis of randomness is accepted. ■

It can be shown that, whennandmare both large andH 0 is true, Rwill have
approximately a normal distribution with mean and standard deviation given by

μ=

2 nm

n+m

+1 and σ=

√

2 nm(2nm−n−m)

(n+m)^2 (n+m−1)

(12.5.2)

Therefore, whennandmare both large

PH 0 {R≤r}=PH 0

{

R−μ

σ

≤

r−μ

σ

}

≈P

{

Z≤

r−μ

σ

}

, Z∼N(0, 1)

=

(

r−μ

σ

)