Page 115 - Statistics and Data Analysis in Geology

Analysis of Sequences of Data

than the expected number of runs from a random arrangement; the null hypothesis
and alternative are

Hi: U>U
and too many runs leads to rejection. The test is one-tailed. Conversely, we may
wish to determine if the sequence contains an improbably low number of runs. The
appropriate alternatives are
Ho: Ur8
H1: u<u

and too few runs will cause rejection of the null hypothesis. Again, the test is one-
tailed. We may wish to reject either form of nonrandomness. A two-tailed test is
appropriate, with hypotheses
Ho: U=8
Hi: Uf8
We can work through the test procedure for the first series of coin flips and
determine the likelihood of achieving this sequence by a random process. The null
hypothesis states that there is no difference between the observed number of runs
and the mean number of runs from random sequences of the same size. We will use
a two-tailed test, and reject if there are too many or too few runs in the sequence.
Therefore, the proper alternative is
Hi: UfU

Using a 5% (a = 0.05) level of significance, our critical regions are bounded by
-1.96 and +1.96. We first calculate the expected mean and standard deviation of
runs for random sequences having nl heads (nl = 11) and n2 tails (n2 = 9):

2 11 * 9)(2 * 11 9 - 11 - 9)
-
aiJ 2-( = 4.6
(9 + 11)*(9 + 11 - 1)
The test statistic is
z=- U-U % 13- 10.9 = 1.0
UU 2.1
The number of runs in the sequence is one standard deviation from the mean of
all runs possible in such a sequence, and does not fall within the critical region.
Therefore, the number of runs does not suggest that the sequence is nonrandom.
The other sequences, in contrast, yield very different test results. Because nl and
nz are the same for all three sequences, 8 and (TU also are the same. For the second
sequence, the test statistic is
2 - 10.9
z= = -4.2
2.1
and for the third,
19 - 10.9
z= = 3.9
2.1
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