How would you test it for independence based on runs above and runs below the mean for the significance level $\alpha$= 0.05 and the critical value Z$_{0.025}$=1.96

1. Define hypothesis for testing independence as

   $H_0: R; \nu$ independently

   $H_1: R; \not{\nu}$ independently

2. The sequence of runs above and below the mean (0.465) are:

   $\text{- - - - + - + - - +}$

   $\text{- - - - + - + - + +}$

3. No. of observations above the mean $n_1 = 7$

   No. of observations below the mean $n_2 = 13$

   Total no. of runs = $b = 12$

4. Mean and variance of b

   $\begin{aligned} H_{b} &=\frac{2 n_{1} n_{2}}{N}+\frac{1}{2} \\ &=\frac{2(7)(13)}{20}+\frac{1}{2} \\ &=20 \cdot 5 \end{aligned}$

   $\begin{aligned} \sigma_{b}^{2} &=\frac{2 n_{1} n_{2}\left(2 n_{1} n_{2}-N\right)}{N^{2}(N-1)} \\ &=\frac{2(7)(13)[2(7)(13)-420]}{20^{2}(20-1)} \\ &=\frac{182\lceil 162]}{400(19)}=3.879 \end{aligned}$

5. Standard normal statistics

   $\begin{aligned} z_{0} &=\frac{b-\mu b}{\sigma b} \\ &=\frac{12-20 \cdot 5}{\sqrt{3 \cdot 879}}=\frac{-8.5}{\sqrt{3 \cdot 879}}=-4 \cdot 315 \end{aligned}$

6. Determine the critical value $Z_{\frac{\alpha}{2}}$ and $-Z_{\frac{\alpha}{2}}$ for specified signifance level $\alpha$ from table A3

   Given as $Z_{0.025} = 1.96$

Since $-4 \cdot 315\lt-Z_{0.025}=1.96$

We say $H_0$ is rejected and random numbers are not independent.