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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

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

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