# 6: Inference for Categorical Data

Chapter 6 introduces inference in the setting of categorical data. We use these methods to answer questions like the following:

- What proportion of the American public approves of the job the Supreme Court is doing?
- The Pew Research Center conducted a poll about support for the 2010 health care law, and they used two forms of the survey question. Each respondent was randomly given one of the two questions. What is the difference in the support for respondents under the two question orderings?

We will find that the methods we learned in previous chapters are very useful in these settings. For example, sample proportions are well characterized by a nearly normal distribution when certain conditions are satisfied, making it possible to employ the usual confidence interval and hypothesis testing tools. In other instances, such as those with contingency tables or when sample size conditions are not met, we will use a different distribution, though the core ideas remain the same.

- 6.1: Inference for a Single Proportion
- A sample proportion can be described as a sample mean.

- 6.2: Difference of Two Proportions
- We would like to make conclusions about the difference in two population proportions: p1−p2. We consider three examples. In the first, we compare the approval of the 2010 healthcare law under two different question phrasings. In the second application, a company weighs whether they should switch to a higher quality parts manufacturer.

- 6.3: Testing for Goodness of Fit using Chi-Square (Special Topic)
- The chi-square distribution is sometimes used to characterize data sets and statistics that are always positive and typically right skewed. Recall the normal distribution had two parameters - mean and standard deviation - that could be used to describe its exact characteristics. The chi-square distribution has just one parameter called degrees of freedom (df), which inuences the shape, center, and spread of the distribution.

- 6.5: Small Sample Hypothesis Testing for a Proportion (Special Topic)
- In this section we develop inferential methods for a single proportion that are appropriate when the sample size is too small to apply the normal models. Just like the methods related to the t distribution, these methods can also be applied to large samples.

- 6.7: Exercises
- Exercises for Chapter 6 of the "OpenIntro Statistics" textmap by Diez, Barr and Çetinkaya-Rundel.

### Contributors

David M Diez (Google/YouTube), Christopher D Barr (Harvard School of Public Health), Mine Çetinkaya-Rundel (Duke University)