Chapter 9.3-9.4 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University Outline Inference for

Chapter 9.3-9.4 - Math3200 Intermediate Probability and...

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Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University
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Outline Inference for count data One-way Two-way Nan Lin, Washington University 2
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One-way table Example: testing the uniformity of random digits Are there systematic runs? Are there cyclic patterns? We test whether all 10 digits occur with equal probability Null hypothesis: the digits follow a uniform distribution on {0,1,2,…,9} Alternative hypothesis: otherwise Data: observed from 100 random digits generated by computer Nan Lin, Washington University 3 Digit 0 1 2 3 4 5 6 7 8 9 Observed count 12 7 12 7 13 13 7 13 6 10
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The general setup Null hypothesis: a discrete random variable ? supported on {1, … , ?} follows a distribution 𝐹 0 𝐹 0 specifies that ? ? = ? = ? ?0 , ? = 1, … , ? Alternative hypothesis: ? ? = ? = ? ? , ? = 1, … , ? , at least one ? ? ≠ ? ?0 Observed data: {𝑛 ? , ? = 1, … , ?} the number of times ? appears in the sample When the total 𝑛 = 𝑛 1 + ⋯ + 𝑛 ? is fixed, distribution of the observed counts are multinomial under the null hypothesis, that is, ? 1 , … , ? ? ∼ ??𝑙??(𝑛, ? 10 , … , ? ?0 ) Nan Lin, Washington University 4
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A chi-square test Intuition : If the null hypothesis is true, the observed count should be closed to what is expected under 𝐻 0 Expected count: ? ? = 𝑛? ?0 , ? = 1, … , ? When n is large, from the CLT, we have 𝑛 ? is normally distributed with mean ? ? , so 𝑛 ? − ? ? 2 is chi-square after some proper adjustment. Null distribution: Under 𝐻 0 , ? ? −? ? 2 ? ? ? ?=1 ∼ 𝜒 ?−1 2 Test: if 𝜒 2 = ? ? −? ? 2 ? ? ? ?=1 > 𝜒 ?−1,𝛼 2 , reject 𝐻 0 Nan Lin, Washington University 5
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Connection to testing proportion Nan Lin, Washington University 6
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Tests of Homogeneity Null hypothesis: ? 0 = ? 1 = ⋯ = ? 9 = 1 10 Alternative hypothesis: otherwise 𝑛 = 100 , 𝛼 = 0.1 𝜒 2 = ? ? −? ? 2 ? ? ? ?=1 = 7.4 < 𝜒 9,0.1 2 = 14.684 ???𝑙?? = ? 𝜒 2 > 7.4 = 0.596 Nan Lin, Washington University 7 Digit 0 1 2 3 4 5 6 7 8 9 𝑛 ?
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