Chi-Square Test -- X 2

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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 -- X2

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
• 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
• 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)
• Kolmogorov-Smirnov
• More powerful
• Can be applied to small samples
• Chi Square
• Large Sample size >50 or 100
• Simpler test
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
• 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
• 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)
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 Freedom
• k – s – 1
• Normal: s=2
• Exponential: s = 1
• Uniform: s = 0
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.
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
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

70

60

50

40

30

20

10

0

10 15 20 25 30 35

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