written 2.8 years ago by
yashbeer
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modified 2.8 years ago

Define hypothesis for testing independence as
$H_0: R; \nu$ independently
$H_1: R; \not{\nu}$ independently
The sequence of runs above and below the mean (0.465) are:
$\text{    +  +   +}$
$\text{    +  +  + +}$
No. of observations above the mean $n_1 = 7$
No. of observations below the mean $n_2 = 13$
Total no. of runs = $b = 12$
Mean and variance of b
$\begin{aligned} H_{b} &=\frac{2 n_{1} n_{2}}{N}+\frac{1}{2} \\ &=\frac{2(7)(13)}{20}+\frac{1}{2} \\ &=20 \cdot 5 \end{aligned}$
$\begin{aligned} \sigma_{b}^{2} &=\frac{2 n_{1} n_{2}\left(2 n_{1} n_{2}N\right)}{N^{2}(N1)} \\ &=\frac{2(7)(13)[2(7)(13)420]}{20^{2}(201)} \\ &=\frac{182\lceil 162]}{400(19)}=3.879 \end{aligned}$
Standard normal statistics
$\begin{aligned} z_{0} &=\frac{b\mu b}{\sigma b} \\ &=\frac{1220 \cdot 5}{\sqrt{3 \cdot 879}}=\frac{8.5}{\sqrt{3 \cdot 879}}=4 \cdot 315 \end{aligned}$
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 $4 \cdot 315\ltZ_{0.025}=1.96$
We say $H_0$ is rejected and random numbers are not independent.