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Test the following random numbers for independence by Poker test.

{0.594, 0.928, 0.515, 0.055, 0.507, 0.351, 0.262, 0.797, 0.788, 0.442, 0.097, 0.798, 0.227, 0.127, 0.474, 0.825, 0.007, 0.182, 0.929, 0.852} Use $\alpha$=0.05, $\chi ^2 _{(0.05,1)}$= 3.84 p

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Poker Test

N = 20

Let us assume 3 digit nos. { 594, 928, 515, ..., 852}

Step 1: Define hypothesis

$H_0: R_i \sim$ independently

$H_a: R_i \not{\sim}$ independently

Step 2: Generate frequency distribution table

Combination
i
Observed freq
$O_i$
Exp. Freq.
$E_i = P \times N$
$\frac{\left(b_{1}-E_{1}\right)^{2}}{E_{i}}$
3 diff digits, 1 10 $0.72 \times 20 = 14.4$ 1.34
3 diff digits, 2 0 $0.01 \times 20 = 0.2$ 3.45
Exactly 1 pair, 3 10 $0.27 \times 20 = 5.4$ 3.45

Step 3: Compute the sample test statistics

\begin{aligned} \chi_{0}^{2} &= \sum_{i=1}^{n} \frac{\left(0_{i}-E_{i}\right)^{2}}{E_{i}}\\ &= 1.34 + 3.45 \\ &= 4.79 \end{aligned}

Step 4: Determine critical value for the specified significance level $\alpha$ with (n-1) diff

$\because \quad \alpha=0.05 \quad n-1 = 2 - 1 = 1$

$\chi_{0.05,1}^2 = 3.84 \ (given)$

Step 5:

$\chi_0^2 = 4.79 \gt chi_{0.05, 1} = 3.84$

$\therefore$ Reject $H_0$

i.e. Random nos are not independent.