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The sequence of numbers 0.63, 0.49, 0.24, 0.89 and 0.71 has been generated.

Use the Komogorov-Smirnov test with $\alpha$= 0.05 to determine if the hypothesis that the numbers are uniformly distributed on the interval[0,1] can be rejected. Use D$_{0.05}$= 0.565

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K-S Test

  1. Define the hypothesis for testing the uniformity

    $H_0: R_i \sim \cup [0,1]$

    $H_i: R_i \not{\sim} \cup [0,1]$

  2. Rank data in increasing order

    $0.24 \le 0.49 \le 0.63 \le 0.71 \le 0.89$

  3. Compute $D^+$ and $D^-$

i 1 2 3 4 5
$R_i$ 0.24 0.49 0.63 0.71 0.89
$\frac{i}{N}$ 0.17 0.33 0.67 0.83 1.00
$\frac{i}{N} - R_i$ - - 0.04 0.12 0.11
$R_{i}-\frac{(i-1)}{N}$ 0.24 0.32 0.13 0.04 0.06

$\therefore D^+ = \text{max {0.04, 0.12, 0.11}} = 0.12$

and $\therefore D^- = \text{max {0.24, 0.32, 0.13, 0.04, 0.06}} = 0.32$

  1. Compute D

    $\begin{aligned} D &=\max \left(D^{+}, D^{-}\right) \\ &=\max (0 \cdot(2,0 \cdot 32)\\ &=0.32 \end{aligned}$

  2. Determine the critical value $D_x$ for specified level of signifance $\alpha = 0.05$ and sample size $N=5$

$$ D_{0.05, 5} = 0.565 \text{ (given) } $$

  1. Since $D=0 \cdot 32 \lt D_{0.05,5}=0.565 \Rightarrow H_{0}$ is not rejected.

    From this, we can say that given set of random numbers are uniformly distributed.

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