Research Methodology - Dr. Krishan K. Pandey

Testing of Hypotheses-II 301
r = number of runs.
In the given case the values of n 1 , n 2 and r would be as follows:
n 1 = 20; n 2 = 10; r = 7
The sampling distribution of ‘r’ statistic, the number of runs, is to be used and this distribution has
its mean
and the standard deviation σ r
=
2nn
1 2
μ r
2nn
1 2 =
n + n
1 2
+ 1
2nn
1 2 − n1 − n2
2 bn1 + n2g bn1 + n2
− 1g
In the given case, we work out the values of μ r and σ r as follows:
bgb 2 20gb10g μ r =
+ 1= 1433 .
20 + 10
and σ r =
bgb 2 20gb10gb2× 20 × 10 − 20 − 10g
= 238 .
2 b20 + 10g b20 + 10 − 1g
For testing the null hypothesis concerning the randomness of the planted trees, we should have been
given the level of significance. Suppose it is 1% or 0.01. Since too many or too few runs would
indicate that the process by which the trees were planted was not random, a two-tailed test is
appropriate which can be indicated as follows on the assumption * that the sampling distribution of r
can be closely approximated by the normal distribution.
Limit
( 258 . ) ( 258 . )
r r
0.005 of area 0.005 of area
0.495 of
area
Fig. 12.4
r r
0.495 of
area
* This assumption can be applied when n1 and n 2 are sufficiently large i.e., they should not be less than 10. But in case
n 1 or n 2 is so small that the normal curve approximation assumption cannot be used, then exact tests may be based on
special tables which can be seen in the book Non-parametric Statistics for the Behavioural Science by S. Siegel.
Limit
8.19 r 14. 33 20.47
(Shaded area shows the
rejection regions)