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1 Chi-Square Test -- X 2 Test of Goodness of Fit

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2 (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

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3 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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4 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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5 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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6 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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7 Chi-Square Steps - #1 Arrange the n observations into k classes Test Statistic: X 2 = Σ (i=0..k) ( O i – E i ) 2 / E i O i = observed # in i th class E i = expected # in i th class Approximates a X 2 distribution with (k-s-1) degrees of freedom

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8 Degrees of Freedom Approximates a X 2 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: X 2 (alpha,dof) from table Ho reject if X 2 > X 2 (alpha,dof)

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9 X 2 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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10 Degrees of Freedom k – s – 1 Normal: s=2 Exponential: s = 1 Uniform: s = 0

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11 X 2 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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12 X 2 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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13 X 2 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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14 Graph of Data 10 15 20 25 30 35 70 60 50 40 30 20 10 0

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15 X 2 Example X 2 Values – (O-E) 2 /E 10-15 = (5-1) 2 /53.2 15-20 = (27-70) 2 /2768.4 20-25 = (68-68) 2 /680 25-30 = (68-41) 2 /6810.7 30-35 = (27-10) 2 /2710.7 35-40+ = (5-10) 2 /45 Total98

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16 X 2 Example DOF = 6-3 = 3 Alpha = 0.05 X 2 table value = 7.81 X 2 calculated = 98 Reject Hypothesis

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Chi Square Test for Goodness of Fit Determining if our sample fits the way it should be.

Chi Square Test for Goodness of Fit Determining if our sample fits the way it should be.

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