Goodness-of-Fit Tests

1. Goodness-of-Fit Tests • Comparing k Population Proportions • If observations from a population can be classified into k categories, a Goodness-of-Fit test can be used to test the null hypothesis that fully specifies these probabilities.

2. Multinomial Distribution • Useful for modeling situations where an experiment has k mutually exclusive and exhaustive outcomes A1,A2, …, Ak with associated probabilities p1,p2, …, pk. • The experiment is repeated n times, and the number of occurrences of each outcome is X1,X2, …, Xk . (Sometimes denoted Oi) • The expected number of outcomes of type i is denoted Ei= npi

3. Comparing k Proportions • Appropriate hypothesis:

4. Chi-Square Goodness-of-Fit • Test Statistic: • Rejection Region always right-tailed • For H1: at least one misspecified pi Reject H0 for c2 ≥ ca2

5. Goodness-of-Fit Tests • Testing Probability Models • Pearson’s Chi-square goodness-of-fit test is useful for checking the validity of discrete and continuous probability models. • For discrete distributions, the probability mass function is used to define each pi, which is used to compute Ei. • For continuous distributions, the data is divided into classes and the probability density function is used to compute the pi, from which Ei is computed.

6. Testing Probability Models • The null hypothesis either specifies the probability distribution being tested, or uses the pmf or pdf of that model to specify the individual probabilities. • Appropriate hypothesis:

7. Contingency Tables • Comparing two or more Multinomial Distributions • Test for independence of two attributes of classification in a single population

8. Goodness-of-Fit Tests • Contingency Tables • Useful for summarizing two or more categorical variables at a time. • Expected number of observations in ith row and jth column of table is

9. Chi-Square Goodness-of-Fit • Test Statistic: