1. Chapter 11 Chi Square Distribution and Its applications

2. What is a Chi Square? Chi Square is another probability distribution (like Normal and Student t) Symbol: χ 2 Picture 0

3. What is a Chi Square? (cont) Features: Mode (i.e. Peak) at n – 2. Degrees of Freedom = n – 2. 0

4. Why χ 2 ? The χ 2 distribution allows you to test the independence of two factors. For instance, what about getting a tattoo and the frequency of Hepatitis C? What if you got the tattoo in a tattoo parlor? What if your cousin Dwayne gave you that great tattoo of Count Chocula ? The Chi Squared test for independence allows you to decide whether the chance of having hepatitis is independent of the tattoo status.

5. Requirements 1. Counted Data Condition: Data must be in counts 2. Randomization Condition: Individuals who have been counted should be a random sample from some population. 3. Expected Cell Frequency Condition: The expected values (which you will calculate) should all be at least 5.

6. What to do? Use a contingency table to find the differences between observed and expected values.

7. What to do? Use a contingency table to find the differences between observed and expected values. d.f. = (R-1)(C-1) where R and C are the number of rows/columns

8. Lets Try one We have counts of 626 individuals categorized according to their “tattoo status” and their “hepatitis status”. Hep CNo Hep C Total Tattoo, Parlor 173552 Tattoo, Elsewhere 85361 No Tattoo 22491513 Total47579626

9. We want to test whether tattoo status and hepatitis status are independent. H O : Tattoo status and hepatitis status are independent H A : Tattoo status and hepatitis status are not independent Check the conditions: 1. Counts of data categorized 2. While you don’t have the information, you can assume that they were selected to avoid biases. 3. We’ll find out when we determine E for each cell

10. Compute E Hep CNo Hep C Total Tattoo, Parlor 173552 Tattoo, Elsewhere 85361 No Tattoo 22491513 Total47579626 3.904 56.420 38.516 48.096 4.580 474.484

11. Notice There are some values below 5 – we’ll continue from here for the sake of the data. What are the degrees of freedom? Find the χ 2 value. Using a table, the value for P( χ 2 >57.91) 2 57.91 Less than.0001

12. Conclusion The P-value is very small, so I reject the null hypothesis and conclude that hepatitis status is not independent of tattoo status.

13. Resources www.microbiologybytes.com/maths/graphics/chi.gif