Chi-Square Test --
2
X
Test of Goodness of Fit
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(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
2
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
3
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)
4
Frequency Tests (Uniformity)
 Kolmogorov-Smirnov
 More powerful
 Can be applied to small samples
 Chi Square
 Large Sample size >50 or 100
 Simpler test
5
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%
15-20 = 13.5%
20-25 = 34%
25-30 = 34%
30-35 = 13.5%
35-40> = 2.5%
5
27
68
68
27
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 =
15-20 =
20-25 =
25-30 =
30-35 =
35-40+ =
 Total
(5-1)2/5
(27-70) 2/27
(68-68) 2/68
(68-41) 2/68
(27-10) 2/27
(5-10) 2/4
3.2
68.4
0
10.7
10.7
5
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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