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...

• Notes
• 25

This preview shows page 1 - 8 out of 25 pages.

Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University

Subscribe to view the full document.

Outline Inference for count data One-way Two-way Nan Lin, Washington University 2
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

Subscribe to view the full document.

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
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

Subscribe to view the full document.

Connection to testing proportion Nan Lin, Washington University 6
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 ?

Subscribe to view the full document.

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes