Take ɑ=0.05 and critical value Z 0.025 =1.96 (0.12, 0.01, 0.23, 0.28, 0.89, 0.31, 0.64, 0.28, 0.33, 0.93).

Step1 :

$H_0$: Ri~ Independently

$H_1$: Ri~is not independently

Step 2 :

Given random numbers=0.12, 0.01, 0.23, 0.28, 0.89, 0.31, 0.64, 0.28, 0.33, 0.93

N= Total numbers of random numbers=10

Step 3:

The sequence of runs up and runs down -, +, +, +, -, +, -, +, +

Total number of runs=R=6

Step 4 :

E(R) = (2N-1)/3

E(R) = (2*10-1)/3

E(R) = 19/3= 6.33

V(R) = (16N-29)/90

V(R) = (16*10-29)/90= 1.456

$Z_0$ = [R - E(R)] / $[V(R)]^{0.5}$

$Z_0$ = [10-6.33] / $[1.456]^{0.5}$ = 3.038

As $Z_0$ lies in the shaded area, $H_0$ is rejected.