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

written 2.6 years ago by | modified 19 months ago by |

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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written 2.6 years ago by | • modified 2.6 years ago |

**K-S Test**

Define the hypothesis for testing the uniformity

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

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

Rank data in increasing order

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

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

Compute D

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

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

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

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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