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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

Chi-Square Test -- X2

Test of Goodness of Fit

pseudo random numbers
(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
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
hypotheses significance level
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)
frequency tests uniformity
Frequency Tests (Uniformity)
  • Kolmogorov-Smirnov
    • More powerful
    • Can be applied to small samples
  • Chi Square
    • Large Sample size >50 or 100
    • Simpler test
overview
Overview
  • Not 100% accurate
  • Formalizes the idea of comparing histograms to candidate probability functions
  • Valid for large samples
  • Valid for Discrete & Continuous
chi square steps 1
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

degrees of freedom
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)
x 2 rules
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
degrees of freedom10
Degrees of Freedom
  • k – s – 1
  • Normal: s=2
  • Exponential: s = 1
  • Uniform: s = 0
x 2 example
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.
x 2 example12
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
x 2 example13
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
graph of data
Graph of Data

70

60

50

40

30

20

10

0

10 15 20 25 30 35

x 2 example15
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
x 2 example16
X2 Example
  • DOF = 6-3 = 3
  • Alpha = 0.05
  • X2 table value = 7.81
  • X2 calculated = 98
  • Reject Hypothesis
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