1. Chapter 16 – Categorical Data Analysis Math 22 Introductory Statistics

2. Chi-Square w Categorical data are statistically analyzed by means of a chi-square statistic. w A single variable is analyzed with the chi- square goodness-of-fit test. w The goodness-of-fit test consists of determining whether the frequency counts in the categories of the variable agree with a specific distribution.

3. The Multinomial Experiment w The experiment consist of n identical experiments. w The outcome of each trial falls into one of k categories.

4. The Multinomial Experiment  The probabilities associated with the k outcomes denoted by  ,  ,  ,…,  k remain the same from trial to trial. Since there are k possible outcome we have:

5. The Multinomial Experiment w The experimenter records the values o 1, o 2,....,o k where o j (j = 1, 2,.....,k) is equal to the number of trials in which the outcome is in category j. w Note:o 1 +o 2 +......+o k = n

6. Chi-Square Goodness-of-Fit Test w Application:Multinomial experiments. w Assumptions: w The experiment satisfies the properties of a multinomial experiment. w No expected cell counts, e j, is less than 1, and no more than 20% of the e j ‘s are less than 5. (This is so the chi-square approximation will be good)

7. Chi-Square Goodness-of-Fit Test w The test is a right-tailed test, where the p- value is found in the chi-square table with k-1 degrees of freedom. Usually the exact value cannot be found, but bounds for it can be found from the closest to the observed value of the chi-square statistic. w Chi-Square Statistic:

8. Chi-Square Test of Independence w Application: Test the independence of the classifying variables Assumptions: w The experiment satisfies the properties of a multinomial experiment. w No expected cell counts, e j, is less than 1, and no more than 20% of the e j ‘s are less than 5. (This is so the chi-square approximation will be good)