# 11: Chi-Square Tests and F-Tests

In previous chapters you saw how to test hypotheses concerning population means and population proportions. The idea of testing hypotheses can be extended to many other situations that involve different parameters and use different test statistics. Whereas the standardized test statistics that appeared in earlier chapters followed either a normal or Student t-distribution, in this chapter the tests will involve two other very common and useful distributions, the chi-square and the F-distributions. The chi-square distribution arises in tests of hypotheses concerning the independence of two random variables and concerning whether a discrete random variable follows a specified distribution. The F-distribution arises in tests of hypotheses concerning whether or not two population variances are equal and concerning whether or not three or more population means are equal.

• 11.1: Chi-Square Tests for Independence
All the chi-square distributions form a family, and each of its members is also specified by a parameter df, the number of degrees of freedom.
• 11.2: Chi-Square One-Sample Goodness-of-Fit Tests
The chi-square goodness-of-fit test can be used to evaluate the hypothesis that a sample is taken from a population with an assumed specific probability distribution.
• 11.3: F-tests for Equality of Two Variances
Another important and useful family of distributions in statistics is the family of F-distributions. An F random variable is a random variable that assumes only positive values and follows an F-distribution. Each member of the F-distribution family is specified by a pair of parameters called degrees of freedom. An F-test can be used to evaluate the hypothesis of two identical normal population variances.
• 11.4: F-Tests in One-Way ANOVA
In this section we will learn to compare three or more population means at the same time, which is often of interest in practical applications. For example, an administrator at a university may be interested in knowing whether student grade point averages are the same for different majors. In another example, an oncologist may be interested in knowing whether patients with the same type of cancer have the same average survival times under several different competing cancer treatments.
• 11.E: Chi-Square Tests and F-Tests (Exercises)
These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang.