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Chi-Square Test -- X 2. Test of Goodness of Fit. (Pseudo) Random Numbers . Uniform : values conform to a uniform distribution Independent : probability of observing a particular value is independent of the previous values Should always test uniformity. Test for Independence.

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Chi square test x 2 l.jpg

Chi-Square Test -- X2

Test of Goodness of Fit


Pseudo random numbers l.jpg
(Pseudo) Random Numbers

  • Uniform: values conform to a uniform distribution

  • Independent: probability of observing a particular value is independent of the previous values

  • Should always test uniformity


Test for independence l.jpg
Test for Independence

  • Autocorrelation Test

    • Tests the correlation between successive numbers and compares to the expected correlation of zero

    • e.g. 2 3 2 3 2 4 2 3

      • There is a correlation between 2 & 3

    • We won’t do this test

      • software available


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Hypotheses & Significance Level

  • Null Hypotheses – Ho

    • Numbers are distributed uniformly

    • Failure to reject Ho shows that evidence of non-uniformity has not been detected

  • Level of Significance – α (alpha)

    • α = P(reject Ho|Ho is true)


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Frequency Tests (Uniformity)

  • Kolmogorov-Smirnov

    • More powerful

    • Can be applied to small samples

  • Chi Square

    • Large Sample size >50 or 100

    • Simpler test


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Overview

  • Not 100% accurate

  • Formalizes the idea of comparing histograms to candidate probability functions

  • Valid for large samples

  • Valid for Discrete & Continuous


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Chi-Square Steps - #1

  • Arrange the n observations into k classes

  • Test Statistic:

    • X2 = Σ(i=0..k) ( Oi – Ei)2 / Ei

    • Oi = observed # in ith class

    • Ei = expected # in ith class

  • Approximates a X2 distribution with

    (k-s-1) degrees of freedom


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Degrees of Freedom

  • Approximates a X2 distribution with (k-s-1) degrees of freedom

  • s = # of parameters for the dist.

  • Ho: RV X conforms to ?? distribution with parameters ??

  • H1: RV X does not conform

  • Critical value: X2(alpha,dof) from table

  • Ho reject if X2 > X2(alpha,dof)


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X2 Rules

  • Each Ei > 5

  • If discrete, each value should be separate group

  • If group too small, can combine adjacent, then reduce dof by 1

  • Suggested values

    • n = 50, k = 5 – 10

    • n = 100, k = 10 – 20

    • n > 100, k = sqrt(n) – n/5


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Degrees of Freedom

  • k – s – 1

  • Normal: s=2

  • Exponential: s = 1

  • Uniform: s = 0


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X2 Example

  • Ho: Ages of MSU students conform to a normal distribution with mean 25 and standard deviation 4.

  • Calculate the expected % for 8 ranges of width 5 from the mean.


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X2 Example

  • Expected percentages & values

    • <10-15 = 2.5% 5

    • 15-20 = 13.5% 27

    • 20-25 = 34% 68

    • 25-30 = 34% 68

    • 30-35 = 13.5% 27

    • 35-40> = 2.5% 5


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X2 Example

  • Consider 200 observations with the following results:

    • 10-15 = 1

    • 15-19 = 70

    • 20-24 = 68

    • 25-29 = 41

    • 30-34 = 10

    • 35-40+ = 10


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Graph of Data

70

60

50

40

30

20

10

0

10 15 20 25 30 35


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X2 Example

  • X2 Values – (O-E)2/E

    • 10-15 = (5-1)2/5 3.2

    • 15-20 = (27-70) 2/27 68.4

    • 20-25 = (68-68) 2/68 0

    • 25-30 = (68-41) 2/68 10.7

    • 30-35 = (27-10) 2/27 10.7

    • 35-40+ = (5-10) 2/4 5

    • Total 98


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X2 Example

  • DOF = 6-3 = 3

  • Alpha = 0.05

  • X2 table value = 7.81

  • X2 calculated = 98

  • Reject Hypothesis


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