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Chapter 11 Study Guide
Inference for Distributions of Categorical Data
This condensed note packet is primarily based off of the AP Statistics Textbook. Please read through the
book and fill this out as needed. It is no substitute for proper reading, but hopefully this study guide will
assist you in your understanding of the material. Do what works for you in terms of homework (I would
advise odd numbered problems + review) and/or quizzes. If you have questions, please feel free to contact
me, Enze Chen, through Facebook, email (echen84), or in person. Good luck!
Introduction
Thus far, we have learned how to conduct a hypothesis test for a population mean and population
proportion (Ch. 9), and we have also compared the proportion of successes for two populations (Ch. 10).
But what if we wanted to look at the distribution for a categorical variable in a population? Does there
exist a statistically significant difference between observed and expected counts? The chi-square test
(read "kai;" Greek symbol χ
2
) allows us to determine whether a hypothesized distribution is valid, the
details of which will be flushed out in this chapter.
Section 11.1 - Chi-Square Goodness-of-Fit Tests
The first of three tests we will learn is the chi-square goodness-of-fit test, which allows us to test the
distribution for a single categorical variable. We demonstrate this using the canonical M&M example.
Example 1:
Mars, Inc. says that the distribution of M&Ms is as follows: 24% Blue, 20% Orange, 16% Green, 14%
Yellow, 13% Red, and 13% Brown. Suppose that the following _____-_____ table gives the data from
Enze's bag of M&Ms.
Color:
Blue
Orange
Green
Yellow
Red
Brown
Total
Count:
9
8
12
15
10
6
60
Does this differ from the stated distribution? Let's look at the proportion of Blue M&Ms, which happens
to be 9/60 = 0.15, while the given is 0.24. Using what we have learned before, we could perform a one-
sample z test for a proportion (Ch. 9) to test the hypotheses
H
0
: ___________

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H
a
: ___________
This is bad. Not only is it inefficient because we would have to test each color, but this method also leads
to multiple (possibly contradicting!
[1]
) comparisons and calculates the probability for subgroups, rather
than selecting a random sample of all 60 candies together. Therefore, we turn to the χ
2
goodness-of-fit
test, analyzing the distribution of color collectively.
Parameter: We wish to analyze the color distribution of M&Ms.
You can just mention the distribution of interest. No need to search for a specific p/μ.
[1] See Appendix I in Study Guide for more information

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Chapter 11 Study Guide Inference for Distributions of Categorical Data This condensed note packet is primarily based off of the AP Statistics Textbook. Please read through the book and fill this out as needed. It is no substitute for proper reading, but hopefully this study guide will assist you in your understanding of the material. Do what works for you in terms of homework (I would advise odd numbered problems + review) and/or quizzes. If you have questions, please feel free to contact me, Enze Chen, through Facebook, email (echen84), or in person. Good luck! Introduction Thus far, we have learned how to conduct a hypothesis test for a population mean and population proportion (Ch. 9), and we have also compared the proportion of successes for two populations (Ch. 10). But what if we wanted to look at the distribution for a categorical variable in a population? Does there exist a statistically significant difference between observed and expected counts? The chi-square test (read "kai;" Greek symbol χ2) allows us to determine whether a hypothesized distribution is valid, the details of which will be flushed out in this chapter. Section 11.1 - Chi-Square Goodness-of-Fit Tests The first of three tests we will learn is the chi-square goodness-of-fit test, which allows us to test the distribution for a single categorical variable. We demonstrate this using the canonical M&M example. Example 1: Mars, Inc. says that the distribution of M&Ms is as follows: 24% Blue, 20% Ora ...
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